universidad de murcia - multiplicatively semiprime algebras · 2012-04-19 · viii encuentro de la...
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Multiplicatively semiprimealgebras
Juan Carlos Cabello PíñarVIII Encuentro de la Red de Análisis Funcional 2012.
La Manga, 19-21 abril 2012
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 1 / 27
Results are contained in
Structure theory for multiplicatively semiprime algebras. J. of Algebra
(2004)Algebras whose multiplication algebra is semiprime. A decompositiontheorem. J. of Algebra (2008).with M. Cabreraε-complemented algebras. J. of algebra (2012),with M. Cabrera and E. Nieto
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 2 / 27
Results are contained inStructure theory for multiplicatively semiprime algebras. J. of Algebra
(2004)
Algebras whose multiplication algebra is semiprime. A decompositiontheorem. J. of Algebra (2008).
with M. Cabreraε-complemented algebras. J. of algebra (2012),with M. Cabrera and E. Nieto
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 2 / 27
Results are contained inStructure theory for multiplicatively semiprime algebras. J. of Algebra
(2004)Algebras whose multiplication algebra is semiprime. A decompositiontheorem. J. of Algebra (2008).
with M. Cabrera
ε-complemented algebras. J. of algebra (2012),with M. Cabrera and E. Nieto
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 2 / 27
Results are contained inStructure theory for multiplicatively semiprime algebras. J. of Algebra
(2004)Algebras whose multiplication algebra is semiprime. A decompositiontheorem. J. of Algebra (2008).
with M. Cabrera
ε-complemented algebras. J. of algebra (2012),with M. Cabrera and E. Nieto
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 2 / 27
Results are contained inStructure theory for multiplicatively semiprime algebras. J. of Algebra
(2004)Algebras whose multiplication algebra is semiprime. A decompositiontheorem. J. of Algebra (2008).with M. Cabrera
ε-complemented algebras. J. of algebra (2012),with M. Cabrera and E. Nieto
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 2 / 27
Results are contained inStructure theory for multiplicatively semiprime algebras. J. of Algebra
(2004)Algebras whose multiplication algebra is semiprime. A decompositiontheorem. J. of Algebra (2008).with M. Cabrera
ε-complemented algebras. J. of algebra (2012),with M. Cabrera and E. Nieto
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 2 / 27
Results are contained inStructure theory for multiplicatively semiprime algebras. J. of Algebra
(2004)Algebras whose multiplication algebra is semiprime. A decompositiontheorem. J. of Algebra (2008).with M. Cabreraε-complemented algebras. J. of algebra (2012),with M. Cabrera and E. Nieto
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 2 / 27
Results are contained inStructure theory for multiplicatively semiprime algebras. J. of Algebra
(2004)Algebras whose multiplication algebra is semiprime. A decompositiontheorem. J. of Algebra (2008).with M. Cabreraε-complemented algebras. J. of algebra (2012),with M. Cabrera and E. Nieto
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 2 / 27
algebraic concepts
Definition
An algebra A is a vector space equipped with a product (a bilinear mapA× A −→ A).
S1, S2 subspaces of an algebra A, S1S2 is the subspace of A generated byall the products xy , for x ∈ S1 and y ∈ S2.
A subspace I of A is said to be an ideal (two-sided) if
IA, AI ⊆ I.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 3 / 27
algebraic concepts
Definition
An algebra A is a vector space equipped with a product (a bilinear mapA× A −→ A).
S1, S2 subspaces of an algebra A, S1S2 is the subspace of A generated byall the products xy , for x ∈ S1 and y ∈ S2.
A subspace I of A is said to be an ideal (two-sided) if
IA, AI ⊆ I.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 3 / 27
algebraic concepts
DefinitionAn algebra A is a vector space equipped with a product (a bilinear mapA× A −→ A).
S1, S2 subspaces of an algebra A, S1S2 is the subspace of A generated byall the products xy , for x ∈ S1 and y ∈ S2.
A subspace I of A is said to be an ideal (two-sided) if
IA, AI ⊆ I.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 3 / 27
algebraic concepts
DefinitionAn algebra A is a vector space equipped with a product (a bilinear mapA× A −→ A).
S1, S2 subspaces of an algebra A, S1S2 is the subspace of A generated byall the products xy , for x ∈ S1 and y ∈ S2.
A subspace I of A is said to be an ideal (two-sided) if
IA, AI ⊆ I.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 3 / 27
algebraic concepts
DefinitionAn algebra A is a vector space equipped with a product (a bilinear mapA× A −→ A).
S1, S2 subspaces of an algebra A, S1S2 is the subspace of A generated byall the products xy , for x ∈ S1 and y ∈ S2.
A subspace I of A is said to be an ideal (two-sided) if
IA, AI ⊆ I.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 3 / 27
algebraic concepts
DefinitionAn algebra A is a vector space equipped with a product (a bilinear mapA× A −→ A).
S1, S2 subspaces of an algebra A, S1S2 is the subspace of A generated byall the products xy , for x ∈ S1 and y ∈ S2.
A subspace I of A is said to be an ideal (two-sided) if
IA, AI ⊆ I.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 3 / 27
algebraic concepts
DefinitionAn algebra A is a vector space equipped with a product (a bilinear mapA× A −→ A).
S1, S2 subspaces of an algebra A, S1S2 is the subspace of A generated byall the products xy , for x ∈ S1 and y ∈ S2.
A subspace I of A is said to be an ideal (two-sided) if
IA, AI ⊆ I.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 3 / 27
algebraic concepts
DefinitionAn algebra A is a vector space equipped with a product (a bilinear mapA× A −→ A).
S1, S2 subspaces of an algebra A, S1S2 is the subspace of A generated byall the products xy , for x ∈ S1 and y ∈ S2.
A subspace I of A is said to be an ideal (two-sided) if
IA, AI ⊆ I.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 3 / 27
algebraic concepts
DefinitionAn algebra A is a vector space equipped with a product (a bilinear mapA× A −→ A).
S1, S2 subspaces of an algebra A, S1S2 is the subspace of A generated byall the products xy , for x ∈ S1 and y ∈ S2.
A subspace I of A is said to be an ideal (two-sided) if
IA, AI ⊆ I.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 3 / 27
DefinitionAn algebra A is associative if (xy)x = x(yz) ∀x , y , z ∈ A.
Examplenot associative algebrasJ. Graves and A. Cayley discovered octonion algebra (-not associative-extension of quaternion algebra).P. Pascal, J. V. Newman y E.Wigner (1934).S. Okubo "Introduction to Octonion and other Non-associative algebras inPhysics" (1995)The algebra of observables (self-adjoints operators) forms a Jordan algebra(nearly associative)
non-associative= not necessarily associative
DefinitionA non-associative algebra. L(A), the algebra of all linear operators on A, isan associative algebra.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 4 / 27
DefinitionAn algebra A is associative if (xy)x = x(yz) ∀x , y , z ∈ A.
Examplenot associative algebras
J. Graves and A. Cayley discovered octonion algebra (-not associative-extension of quaternion algebra).P. Pascal, J. V. Newman y E.Wigner (1934).S. Okubo "Introduction to Octonion and other Non-associative algebras inPhysics" (1995)The algebra of observables (self-adjoints operators) forms a Jordan algebra(nearly associative)
non-associative= not necessarily associative
DefinitionA non-associative algebra. L(A), the algebra of all linear operators on A, isan associative algebra.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 4 / 27
DefinitionAn algebra A is associative if (xy)x = x(yz) ∀x , y , z ∈ A.
Examplenot associative algebrasJ. Graves and A. Cayley discovered octonion algebra (-not associative-extension of quaternion algebra).
P. Pascal, J. V. Newman y E.Wigner (1934).S. Okubo "Introduction to Octonion and other Non-associative algebras inPhysics" (1995)The algebra of observables (self-adjoints operators) forms a Jordan algebra(nearly associative)
non-associative= not necessarily associative
DefinitionA non-associative algebra. L(A), the algebra of all linear operators on A, isan associative algebra.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 4 / 27
DefinitionAn algebra A is associative if (xy)x = x(yz) ∀x , y , z ∈ A.
Examplenot associative algebrasJ. Graves and A. Cayley discovered octonion algebra (-not associative-extension of quaternion algebra).P. Pascal, J. V. Newman y E.Wigner (1934).
S. Okubo "Introduction to Octonion and other Non-associative algebras inPhysics" (1995)The algebra of observables (self-adjoints operators) forms a Jordan algebra(nearly associative)
non-associative= not necessarily associative
DefinitionA non-associative algebra. L(A), the algebra of all linear operators on A, isan associative algebra.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 4 / 27
DefinitionAn algebra A is associative if (xy)x = x(yz) ∀x , y , z ∈ A.
Examplenot associative algebrasJ. Graves and A. Cayley discovered octonion algebra (-not associative-extension of quaternion algebra).P. Pascal, J. V. Newman y E.Wigner (1934).S. Okubo "Introduction to Octonion and other Non-associative algebras inPhysics" (1995)
The algebra of observables (self-adjoints operators) forms a Jordan algebra(nearly associative)
non-associative= not necessarily associative
DefinitionA non-associative algebra. L(A), the algebra of all linear operators on A, isan associative algebra.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 4 / 27
DefinitionAn algebra A is associative if (xy)x = x(yz) ∀x , y , z ∈ A.
Examplenot associative algebrasJ. Graves and A. Cayley discovered octonion algebra (-not associative-extension of quaternion algebra).P. Pascal, J. V. Newman y E.Wigner (1934).S. Okubo "Introduction to Octonion and other Non-associative algebras inPhysics" (1995)The algebra of observables (self-adjoints operators) forms a Jordan algebra(nearly associative)
non-associative= not necessarily associative
DefinitionA non-associative algebra. L(A), the algebra of all linear operators on A, isan associative algebra.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 4 / 27
DefinitionAn algebra A is associative if (xy)x = x(yz) ∀x , y , z ∈ A.
Examplenot associative algebrasJ. Graves and A. Cayley discovered octonion algebra (-not associative-extension of quaternion algebra).P. Pascal, J. V. Newman y E.Wigner (1934).S. Okubo "Introduction to Octonion and other Non-associative algebras inPhysics" (1995)The algebra of observables (self-adjoints operators) forms a Jordan algebra(nearly associative)
non-associative= not necessarily associative
DefinitionA non-associative algebra. L(A), the algebra of all linear operators on A, isan associative algebra.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 4 / 27
DefinitionAn algebra A is associative if (xy)x = x(yz) ∀x , y , z ∈ A.
Examplenot associative algebrasJ. Graves and A. Cayley discovered octonion algebra (-not associative-extension of quaternion algebra).P. Pascal, J. V. Newman y E.Wigner (1934).S. Okubo "Introduction to Octonion and other Non-associative algebras inPhysics" (1995)The algebra of observables (self-adjoints operators) forms a Jordan algebra(nearly associative)
non-associative= not necessarily associative
DefinitionA non-associative algebra.
L(A), the algebra of all linear operators on A, isan associative algebra.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 4 / 27
DefinitionAn algebra A is associative if (xy)x = x(yz) ∀x , y , z ∈ A.
Examplenot associative algebrasJ. Graves and A. Cayley discovered octonion algebra (-not associative-extension of quaternion algebra).P. Pascal, J. V. Newman y E.Wigner (1934).S. Okubo "Introduction to Octonion and other Non-associative algebras inPhysics" (1995)The algebra of observables (self-adjoints operators) forms a Jordan algebra(nearly associative)
non-associative= not necessarily associative
DefinitionA non-associative algebra. L(A),
the algebra of all linear operators on A, isan associative algebra.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 4 / 27
DefinitionAn algebra A is associative if (xy)x = x(yz) ∀x , y , z ∈ A.
Examplenot associative algebrasJ. Graves and A. Cayley discovered octonion algebra (-not associative-extension of quaternion algebra).P. Pascal, J. V. Newman y E.Wigner (1934).S. Okubo "Introduction to Octonion and other Non-associative algebras inPhysics" (1995)The algebra of observables (self-adjoints operators) forms a Jordan algebra(nearly associative)
non-associative= not necessarily associative
DefinitionA non-associative algebra. L(A), the algebra of all linear operators on A, isan associative algebra.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 4 / 27
Are there any subalgebra of L(A) whose structure is closely related to thealgebra structure of A?The answer is yes. The multiplication algebra
Definitiona ∈ A,The operators of left and right multiplication La and Ra by a on A,La, Ra : A −→ A are defined by
La(x) = ax and Ra(x) = xa.
The multiplication algebra M(A) of A is defined as the subalgebra of L(A)generated by the identity operator IdA and the set La, Ra : a ∈ A.
Is there a vehicle of information back and forth between the twoalgebras? The answer is yes. The ε-closure.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 5 / 27
Are there any subalgebra of L(A) whose structure is closely related to thealgebra structure of A?
The answer is yes. The multiplication algebra
Definitiona ∈ A,The operators of left and right multiplication La and Ra by a on A,La, Ra : A −→ A are defined by
La(x) = ax and Ra(x) = xa.
The multiplication algebra M(A) of A is defined as the subalgebra of L(A)generated by the identity operator IdA and the set La, Ra : a ∈ A.
Is there a vehicle of information back and forth between the twoalgebras? The answer is yes. The ε-closure.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 5 / 27
Are there any subalgebra of L(A) whose structure is closely related to thealgebra structure of A?
The answer is yes. The multiplication algebra
Definitiona ∈ A,The operators of left and right multiplication La and Ra by a on A,La, Ra : A −→ A are defined by
La(x) = ax and Ra(x) = xa.
The multiplication algebra M(A) of A is defined as the subalgebra of L(A)generated by the identity operator IdA and the set La, Ra : a ∈ A.
Is there a vehicle of information back and forth between the twoalgebras? The answer is yes. The ε-closure.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 5 / 27
Are there any subalgebra of L(A) whose structure is closely related to thealgebra structure of A?The answer is yes. The multiplication algebra
Definitiona ∈ A,The operators of left and right multiplication La and Ra by a on A,La, Ra : A −→ A are defined by
La(x) = ax and Ra(x) = xa.
The multiplication algebra M(A) of A is defined as the subalgebra of L(A)generated by the identity operator IdA and the set La, Ra : a ∈ A.
Is there a vehicle of information back and forth between the twoalgebras? The answer is yes. The ε-closure.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 5 / 27
Are there any subalgebra of L(A) whose structure is closely related to thealgebra structure of A?The answer is yes. The multiplication algebra
Definitiona ∈ A,The operators of left and right multiplication La and Ra by a on A,La, Ra : A −→ A are defined by
La(x) = ax and Ra(x) = xa.
The multiplication algebra M(A) of A is defined as the subalgebra of L(A)generated by the identity operator IdA and the set La, Ra : a ∈ A.
Is there a vehicle of information back and forth between the twoalgebras? The answer is yes. The ε-closure.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 5 / 27
Are there any subalgebra of L(A) whose structure is closely related to thealgebra structure of A?The answer is yes. The multiplication algebra
Definitiona ∈ A,The operators of left and right multiplication La and Ra by a on A,La, Ra : A −→ A are defined by
La(x) = ax and Ra(x) = xa.
The multiplication algebra M(A) of A is defined as the subalgebra of L(A)generated by the identity operator IdA and the set La, Ra : a ∈ A.
Is there a vehicle of information back and forth between the twoalgebras? The answer is yes. The ε-closure.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 5 / 27
Are there any subalgebra of L(A) whose structure is closely related to thealgebra structure of A?The answer is yes. The multiplication algebra
Definitiona ∈ A,The operators of left and right multiplication La and Ra by a on A,La, Ra : A −→ A are defined by
La(x) = ax and Ra(x) = xa.
The multiplication algebra M(A) of A is defined as the subalgebra of L(A)generated by the identity operator IdA and the set La, Ra : a ∈ A.
Is there a vehicle of information back and forth between the twoalgebras?
The answer is yes. The ε-closure.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 5 / 27
Are there any subalgebra of L(A) whose structure is closely related to thealgebra structure of A?The answer is yes. The multiplication algebra
Definitiona ∈ A,The operators of left and right multiplication La and Ra by a on A,La, Ra : A −→ A are defined by
La(x) = ax and Ra(x) = xa.
The multiplication algebra M(A) of A is defined as the subalgebra of L(A)generated by the identity operator IdA and the set La, Ra : a ∈ A.
Is there a vehicle of information back and forth between the twoalgebras?
The answer is yes. The ε-closure.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 5 / 27
Are there any subalgebra of L(A) whose structure is closely related to thealgebra structure of A?The answer is yes. The multiplication algebra
Definitiona ∈ A,The operators of left and right multiplication La and Ra by a on A,La, Ra : A −→ A are defined by
La(x) = ax and Ra(x) = xa.
The multiplication algebra M(A) of A is defined as the subalgebra of L(A)generated by the identity operator IdA and the set La, Ra : a ∈ A.
Is there a vehicle of information back and forth between the twoalgebras? The answer is yes.
The ε-closure.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 5 / 27
Are there any subalgebra of L(A) whose structure is closely related to thealgebra structure of A?The answer is yes. The multiplication algebra
Definitiona ∈ A,The operators of left and right multiplication La and Ra by a on A,La, Ra : A −→ A are defined by
La(x) = ax and Ra(x) = xa.
The multiplication algebra M(A) of A is defined as the subalgebra of L(A)generated by the identity operator IdA and the set La, Ra : a ∈ A.
Is there a vehicle of information back and forth between the twoalgebras? The answer is yes. The ε-closure.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 5 / 27
DefinitionA complete lattice is a partially ordered set L in which all its subsetsxi ; i ∈ Λ have both a supremum, ∨i xi , and an infimum, ∧i xi ).
Every complete lattice has a greatest,1L, and a least element, 0L.
ExampleX vector space,SX = subspaces of X is a complete lattice with
∨ =
, ∧ = ∩, and 1SX= X and 0SX
= 0.
ExampleA algebra, IA = ideals ofA.IA, is also a complete lattice with the same operations.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 6 / 27
DefinitionA complete lattice is a partially ordered set L in which all its subsetsxi ; i ∈ Λ have both a supremum, ∨i xi , and an infimum, ∧i xi ).
Every complete lattice has a greatest,1L, and a least element, 0L.
ExampleX vector space,SX = subspaces of X is a complete lattice with
∨ =
, ∧ = ∩, and 1SX= X and 0SX
= 0.
ExampleA algebra, IA = ideals ofA.IA, is also a complete lattice with the same operations.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 6 / 27
DefinitionA complete lattice is a partially ordered set L in which all its subsetsxi ; i ∈ Λ have both a supremum, ∨i xi , and an infimum, ∧i xi ).
Every complete lattice has a greatest,1L, and a least element, 0L.
Example
X vector space,SX = subspaces of X is a complete lattice with
∨ =
, ∧ = ∩, and 1SX= X and 0SX
= 0.
ExampleA algebra, IA = ideals ofA.IA, is also a complete lattice with the same operations.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 6 / 27
DefinitionA complete lattice is a partially ordered set L in which all its subsetsxi ; i ∈ Λ have both a supremum, ∨i xi , and an infimum, ∧i xi ).
Every complete lattice has a greatest,1L, and a least element, 0L.
ExampleX vector space,SX = subspaces of X is a complete lattice with
∨ =
, ∧ = ∩, and 1SX= X and 0SX
= 0.
ExampleA algebra, IA = ideals ofA.IA, is also a complete lattice with the same operations.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 6 / 27
DefinitionA complete lattice is a partially ordered set L in which all its subsetsxi ; i ∈ Λ have both a supremum, ∨i xi , and an infimum, ∧i xi ).
Every complete lattice has a greatest,1L, and a least element, 0L.
ExampleX vector space,SX = subspaces of X is a complete lattice with
∨ =
, ∧ = ∩, and 1SX= X and 0SX
= 0.
ExampleA algebra, IA = ideals ofA.IA, is also a complete lattice with the same operations.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 6 / 27
DefinitionA complete lattice is a partially ordered set L in which all its subsetsxi ; i ∈ Λ have both a supremum, ∨i xi , and an infimum, ∧i xi ).
Every complete lattice has a greatest,1L, and a least element, 0L.
ExampleX vector space,SX = subspaces of X is a complete lattice with
∨ =
, ∧ = ∩, and 1SX= X and 0SX
= 0.
ExampleA algebra, IA = ideals ofA.
IA, is also a complete lattice with the same operations.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 6 / 27
DefinitionA complete lattice is a partially ordered set L in which all its subsetsxi ; i ∈ Λ have both a supremum, ∨i xi , and an infimum, ∧i xi ).
Every complete lattice has a greatest,1L, and a least element, 0L.
ExampleX vector space,SX = subspaces of X is a complete lattice with
∨ =
, ∧ = ∩, and 1SX= X and 0SX
= 0.
ExampleA algebra, IA = ideals ofA.IA, is also a complete lattice with the same operations.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 6 / 27
Closure Operations
DefinitionA map x → x from a complete lattice L into itself is called a closureoperation if:
x1 ≤ x2 ⇒ x1 ≤ x2, x ≤ x , ∀x , x1, x2 ∈ L,(∧xi)∼ = ∧xi , for every subset xi of L.
x ∈ L, x is ∼-closed (resp, ∼-dense) whenever x = x , (resp. x = 1L).
The set L∼ of all ∼-closed elements of L is a complete lattice for thesupremum and infimum given by xi = (∨xi)∼ and xi = ∧xi .
ExampleX normed space, The norm-closure is a closure operation in SX .The set of all ||.||-closed subspaces of X , S
||.||X
, is a complete lattice for the
supremum and infimum given by ∨Si = (
Si)||.||
and ∧ Si = ∩Si
X and 0 are respectively the largest and the smallest elements in S||.||X
.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 7 / 27
Closure Operations
Definition
A map x → x from a complete lattice L into itself is called a closureoperation if:
x1 ≤ x2 ⇒ x1 ≤ x2, x ≤ x , ∀x , x1, x2 ∈ L,(∧xi)∼ = ∧xi , for every subset xi of L.
x ∈ L, x is ∼-closed (resp, ∼-dense) whenever x = x , (resp. x = 1L).
The set L∼ of all ∼-closed elements of L is a complete lattice for thesupremum and infimum given by xi = (∨xi)∼ and xi = ∧xi .
ExampleX normed space, The norm-closure is a closure operation in SX .The set of all ||.||-closed subspaces of X , S
||.||X
, is a complete lattice for the
supremum and infimum given by ∨Si = (
Si)||.||
and ∧ Si = ∩Si
X and 0 are respectively the largest and the smallest elements in S||.||X
.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 7 / 27
Closure Operations
DefinitionA map x → x from a complete lattice L into itself is called a closureoperation if:
x1 ≤ x2 ⇒ x1 ≤ x2, x ≤ x , ∀x , x1, x2 ∈ L,(∧xi)∼ = ∧xi , for every subset xi of L.
x ∈ L, x is ∼-closed (resp, ∼-dense) whenever x = x , (resp. x = 1L).
The set L∼ of all ∼-closed elements of L is a complete lattice for thesupremum and infimum given by xi = (∨xi)∼ and xi = ∧xi .
ExampleX normed space, The norm-closure is a closure operation in SX .The set of all ||.||-closed subspaces of X , S
||.||X
, is a complete lattice for the
supremum and infimum given by ∨Si = (
Si)||.||
and ∧ Si = ∩Si
X and 0 are respectively the largest and the smallest elements in S||.||X
.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 7 / 27
Closure Operations
DefinitionA map x → x from a complete lattice L into itself is called a closureoperation if:
x1 ≤ x2 ⇒ x1 ≤ x2,
x ≤ x , ∀x , x1, x2 ∈ L,
(∧xi)∼ = ∧xi , for every subset xi of L.
x ∈ L, x is ∼-closed (resp, ∼-dense) whenever x = x , (resp. x = 1L).
The set L∼ of all ∼-closed elements of L is a complete lattice for thesupremum and infimum given by xi = (∨xi)∼ and xi = ∧xi .
ExampleX normed space, The norm-closure is a closure operation in SX .The set of all ||.||-closed subspaces of X , S
||.||X
, is a complete lattice for the
supremum and infimum given by ∨Si = (
Si)||.||
and ∧ Si = ∩Si
X and 0 are respectively the largest and the smallest elements in S||.||X
.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 7 / 27
Closure Operations
DefinitionA map x → x from a complete lattice L into itself is called a closureoperation if:
x1 ≤ x2 ⇒ x1 ≤ x2,
x ≤ x , ∀x , x1, x2 ∈ L,
(∧xi)∼ = ∧xi , for every subset xi of L.
x ∈ L, x is ∼-closed (resp, ∼-dense) whenever x = x , (resp. x = 1L).
The set L∼ of all ∼-closed elements of L is a complete lattice for thesupremum and infimum given by xi = (∨xi)∼ and xi = ∧xi .
ExampleX normed space, The norm-closure is a closure operation in SX .The set of all ||.||-closed subspaces of X , S
||.||X
, is a complete lattice for the
supremum and infimum given by ∨Si = (
Si)||.||
and ∧ Si = ∩Si
X and 0 are respectively the largest and the smallest elements in S||.||X
.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 7 / 27
Closure Operations
DefinitionA map x → x from a complete lattice L into itself is called a closureoperation if:
x1 ≤ x2 ⇒ x1 ≤ x2, x ≤ x , ∀x , x1, x2 ∈ L,
(∧xi)∼ = ∧xi , for every subset xi of L.
x ∈ L, x is ∼-closed (resp, ∼-dense) whenever x = x , (resp. x = 1L).
The set L∼ of all ∼-closed elements of L is a complete lattice for thesupremum and infimum given by xi = (∨xi)∼ and xi = ∧xi .
ExampleX normed space, The norm-closure is a closure operation in SX .The set of all ||.||-closed subspaces of X , S
||.||X
, is a complete lattice for the
supremum and infimum given by ∨Si = (
Si)||.||
and ∧ Si = ∩Si
X and 0 are respectively the largest and the smallest elements in S||.||X
.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 7 / 27
Closure Operations
DefinitionA map x → x from a complete lattice L into itself is called a closureoperation if:
x1 ≤ x2 ⇒ x1 ≤ x2, x ≤ x , ∀x , x1, x2 ∈ L,
(∧xi)∼ = ∧xi , for every subset xi of L.
x ∈ L, x is ∼-closed (resp, ∼-dense) whenever x = x , (resp. x = 1L).
The set L∼ of all ∼-closed elements of L is a complete lattice for thesupremum and infimum given by xi = (∨xi)∼ and xi = ∧xi .
ExampleX normed space, The norm-closure is a closure operation in SX .The set of all ||.||-closed subspaces of X , S
||.||X
, is a complete lattice for the
supremum and infimum given by ∨Si = (
Si)||.||
and ∧ Si = ∩Si
X and 0 are respectively the largest and the smallest elements in S||.||X
.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 7 / 27
Closure Operations
DefinitionA map x → x from a complete lattice L into itself is called a closureoperation if:
x1 ≤ x2 ⇒ x1 ≤ x2, x ≤ x , ∀x , x1, x2 ∈ L,(∧xi)∼ = ∧xi , for every subset xi of L.
x ∈ L, x is ∼-closed (resp, ∼-dense) whenever x = x , (resp. x = 1L).
The set L∼ of all ∼-closed elements of L is a complete lattice for thesupremum and infimum given by xi = (∨xi)∼ and xi = ∧xi .
ExampleX normed space, The norm-closure is a closure operation in SX .The set of all ||.||-closed subspaces of X , S
||.||X
, is a complete lattice for the
supremum and infimum given by ∨Si = (
Si)||.||
and ∧ Si = ∩Si
X and 0 are respectively the largest and the smallest elements in S||.||X
.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 7 / 27
Closure Operations
DefinitionA map x → x from a complete lattice L into itself is called a closureoperation if:
x1 ≤ x2 ⇒ x1 ≤ x2, x ≤ x , ∀x , x1, x2 ∈ L,(∧xi)∼ = ∧xi , for every subset xi of L.
x ∈ L, x is ∼-closed (resp, ∼-dense) whenever x = x , (resp. x = 1L).
The set L∼ of all ∼-closed elements of L is a complete lattice for thesupremum and infimum given by xi = (∨xi)∼ and xi = ∧xi .
ExampleX normed space, The norm-closure is a closure operation in SX .The set of all ||.||-closed subspaces of X , S
||.||X
, is a complete lattice for the
supremum and infimum given by ∨Si = (
Si)||.||
and ∧ Si = ∩Si
X and 0 are respectively the largest and the smallest elements in S||.||X
.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 7 / 27
Closure Operations
DefinitionA map x → x from a complete lattice L into itself is called a closureoperation if:
x1 ≤ x2 ⇒ x1 ≤ x2, x ≤ x , ∀x , x1, x2 ∈ L,(∧xi)∼ = ∧xi , for every subset xi of L.
x ∈ L, x is ∼-closed (resp, ∼-dense) whenever x = x , (resp. x = 1L).
The set L∼ of all ∼-closed elements of L is a complete lattice for thesupremum and infimum given by xi = (∨xi)∼ and xi = ∧xi .
ExampleX normed space, The norm-closure is a closure operation in SX .The set of all ||.||-closed subspaces of X , S
||.||X
, is a complete lattice for the
supremum and infimum given by ∨Si = (
Si)||.||
and ∧ Si = ∩Si
X and 0 are respectively the largest and the smallest elements in S||.||X
.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 7 / 27
Closure Operations
DefinitionA map x → x from a complete lattice L into itself is called a closureoperation if:
x1 ≤ x2 ⇒ x1 ≤ x2, x ≤ x , ∀x , x1, x2 ∈ L,(∧xi)∼ = ∧xi , for every subset xi of L.
x ∈ L, x is ∼-closed (resp, ∼-dense) whenever x = x , (resp. x = 1L).
The set L∼ of all ∼-closed elements of L is a complete lattice for thesupremum and infimum given by xi = (∨xi)∼ and xi = ∧xi .
Example
X normed space, The norm-closure is a closure operation in SX .The set of all ||.||-closed subspaces of X , S
||.||X
, is a complete lattice for the
supremum and infimum given by ∨Si = (
Si)||.||
and ∧ Si = ∩Si
X and 0 are respectively the largest and the smallest elements in S||.||X
.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 7 / 27
Closure Operations
DefinitionA map x → x from a complete lattice L into itself is called a closureoperation if:
x1 ≤ x2 ⇒ x1 ≤ x2, x ≤ x , ∀x , x1, x2 ∈ L,(∧xi)∼ = ∧xi , for every subset xi of L.
x ∈ L, x is ∼-closed (resp, ∼-dense) whenever x = x , (resp. x = 1L).
The set L∼ of all ∼-closed elements of L is a complete lattice for thesupremum and infimum given by xi = (∨xi)∼ and xi = ∧xi .
ExampleX normed space,
The norm-closure is a closure operation in SX .The set of all ||.||-closed subspaces of X , S
||.||X
, is a complete lattice for the
supremum and infimum given by ∨Si = (
Si)||.||
and ∧ Si = ∩Si
X and 0 are respectively the largest and the smallest elements in S||.||X
.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 7 / 27
Closure Operations
DefinitionA map x → x from a complete lattice L into itself is called a closureoperation if:
x1 ≤ x2 ⇒ x1 ≤ x2, x ≤ x , ∀x , x1, x2 ∈ L,(∧xi)∼ = ∧xi , for every subset xi of L.
x ∈ L, x is ∼-closed (resp, ∼-dense) whenever x = x , (resp. x = 1L).
The set L∼ of all ∼-closed elements of L is a complete lattice for thesupremum and infimum given by xi = (∨xi)∼ and xi = ∧xi .
ExampleX normed space, The norm-closure is a closure operation in SX .
The set of all ||.||-closed subspaces of X , S||.||X
, is a complete lattice for the
supremum and infimum given by ∨Si = (
Si)||.||
and ∧ Si = ∩Si
X and 0 are respectively the largest and the smallest elements in S||.||X
.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 7 / 27
Closure Operations
DefinitionA map x → x from a complete lattice L into itself is called a closureoperation if:
x1 ≤ x2 ⇒ x1 ≤ x2, x ≤ x , ∀x , x1, x2 ∈ L,(∧xi)∼ = ∧xi , for every subset xi of L.
x ∈ L, x is ∼-closed (resp, ∼-dense) whenever x = x , (resp. x = 1L).
The set L∼ of all ∼-closed elements of L is a complete lattice for thesupremum and infimum given by xi = (∨xi)∼ and xi = ∧xi .
ExampleX normed space, The norm-closure is a closure operation in SX .The set of all ||.||-closed subspaces of X , S
||.||X
, is a complete lattice
for the
supremum and infimum given by ∨Si = (
Si)||.||
and ∧ Si = ∩Si
X and 0 are respectively the largest and the smallest elements in S||.||X
.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 7 / 27
Closure Operations
DefinitionA map x → x from a complete lattice L into itself is called a closureoperation if:
x1 ≤ x2 ⇒ x1 ≤ x2, x ≤ x , ∀x , x1, x2 ∈ L,(∧xi)∼ = ∧xi , for every subset xi of L.
x ∈ L, x is ∼-closed (resp, ∼-dense) whenever x = x , (resp. x = 1L).
The set L∼ of all ∼-closed elements of L is a complete lattice for thesupremum and infimum given by xi = (∨xi)∼ and xi = ∧xi .
ExampleX normed space, The norm-closure is a closure operation in SX .The set of all ||.||-closed subspaces of X , S
||.||X
, is a complete lattice for the
supremum and infimum given by ∨Si = (
Si)||.||
and ∧ Si = ∩Si
X and 0 are respectively the largest and the smallest elements in S||.||X
.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 7 / 27
Closure Operations
DefinitionA map x → x from a complete lattice L into itself is called a closureoperation if:
x1 ≤ x2 ⇒ x1 ≤ x2, x ≤ x , ∀x , x1, x2 ∈ L,(∧xi)∼ = ∧xi , for every subset xi of L.
x ∈ L, x is ∼-closed (resp, ∼-dense) whenever x = x , (resp. x = 1L).
The set L∼ of all ∼-closed elements of L is a complete lattice for thesupremum and infimum given by xi = (∨xi)∼ and xi = ∧xi .
ExampleX normed space, The norm-closure is a closure operation in SX .The set of all ||.||-closed subspaces of X , S
||.||X
, is a complete lattice for the
supremum and infimum given by ∨Si = (
Si)||.||
and ∧ Si = ∩Si
X and 0 are respectively the largest and the smallest elements in S||.||X
.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 7 / 27
Galois connexions
DefinitionA Galois connexion between two complete lattices L and M is a pair of maps
L
M,
satisfying:
x1 ≤ x2 ⇒ x∗2 ≤ x∗1 and y1 ≤ y2 ⇒ y2 ≤ y1 , for all x1, x2 ∈ L, y1, y2 ∈ M.x ≤ x∗ and y ≤ y∗, for all x ∈ L, y ∈ M.(∨xi)∗ = ∧x∗
iand (∨yi) = ∧y
i, for all subsets xi ⊆ L, yi ⊆ M.
(0L)∗ = 1M and (0M) = 1L.
L
M Galois connexion.
The maps ε : x → x∗ and ε : y → y∗ are closure operations in L and M.
Lε = ε-closed ideals of L) and Mε = ε-closed ideals of M are complete
lattice, and Lε
Mε is an order-reversing bijection pairing.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 8 / 27
Galois connexions
Definition
A Galois connexion between two complete lattices L and M is a pair of mapsL
M,
satisfying:
x1 ≤ x2 ⇒ x∗2 ≤ x∗1 and y1 ≤ y2 ⇒ y2 ≤ y1 , for all x1, x2 ∈ L, y1, y2 ∈ M.x ≤ x∗ and y ≤ y∗, for all x ∈ L, y ∈ M.(∨xi)∗ = ∧x∗
iand (∨yi) = ∧y
i, for all subsets xi ⊆ L, yi ⊆ M.
(0L)∗ = 1M and (0M) = 1L.
L
M Galois connexion.
The maps ε : x → x∗ and ε : y → y∗ are closure operations in L and M.
Lε = ε-closed ideals of L) and Mε = ε-closed ideals of M are complete
lattice, and Lε
Mε is an order-reversing bijection pairing.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 8 / 27
Galois connexions
DefinitionA Galois connexion between two complete lattices L and M
is a pair of mapsL
M,
satisfying:
x1 ≤ x2 ⇒ x∗2 ≤ x∗1 and y1 ≤ y2 ⇒ y2 ≤ y1 , for all x1, x2 ∈ L, y1, y2 ∈ M.x ≤ x∗ and y ≤ y∗, for all x ∈ L, y ∈ M.(∨xi)∗ = ∧x∗
iand (∨yi) = ∧y
i, for all subsets xi ⊆ L, yi ⊆ M.
(0L)∗ = 1M and (0M) = 1L.
L
M Galois connexion.
The maps ε : x → x∗ and ε : y → y∗ are closure operations in L and M.
Lε = ε-closed ideals of L) and Mε = ε-closed ideals of M are complete
lattice, and Lε
Mε is an order-reversing bijection pairing.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 8 / 27
Galois connexions
DefinitionA Galois connexion between two complete lattices L and M is a pair of maps
L
M,
satisfying:
x1 ≤ x2 ⇒ x∗2 ≤ x∗1 and y1 ≤ y2 ⇒ y2 ≤ y1 , for all x1, x2 ∈ L, y1, y2 ∈ M.x ≤ x∗ and y ≤ y∗, for all x ∈ L, y ∈ M.(∨xi)∗ = ∧x∗
iand (∨yi) = ∧y
i, for all subsets xi ⊆ L, yi ⊆ M.
(0L)∗ = 1M and (0M) = 1L.
L
M Galois connexion.
The maps ε : x → x∗ and ε : y → y∗ are closure operations in L and M.
Lε = ε-closed ideals of L) and Mε = ε-closed ideals of M are complete
lattice, and Lε
Mε is an order-reversing bijection pairing.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 8 / 27
Galois connexions
DefinitionA Galois connexion between two complete lattices L and M is a pair of maps
L
M,
satisfying:
x1 ≤ x2 ⇒ x∗2 ≤ x∗1 and y1 ≤ y2 ⇒ y2 ≤ y1 , for all x1, x2 ∈ L, y1, y2 ∈ M.x ≤ x∗ and y ≤ y∗, for all x ∈ L, y ∈ M.(∨xi)∗ = ∧x∗
iand (∨yi) = ∧y
i, for all subsets xi ⊆ L, yi ⊆ M.
(0L)∗ = 1M and (0M) = 1L.
L
M Galois connexion.
The maps ε : x → x∗ and ε : y → y∗ are closure operations in L and M.
Lε = ε-closed ideals of L) and Mε = ε-closed ideals of M are complete
lattice, and Lε
Mε is an order-reversing bijection pairing.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 8 / 27
Galois connexions
DefinitionA Galois connexion between two complete lattices L and M is a pair of maps
L
M,
satisfying:x1 ≤ x2 ⇒ x∗2 ≤ x∗1 and y1 ≤ y2 ⇒ y2 ≤ y1 , for all x1, x2 ∈ L, y1, y2 ∈ M.
x ≤ x∗ and y ≤ y∗, for all x ∈ L, y ∈ M.
(∨xi)∗ = ∧x∗i
and (∨yi) = ∧yi
, for all subsets xi ⊆ L, yi ⊆ M.(0L)∗ = 1M and (0M) = 1L.
L
M Galois connexion.
The maps ε : x → x∗ and ε : y → y∗ are closure operations in L and M.
Lε = ε-closed ideals of L) and Mε = ε-closed ideals of M are complete
lattice, and Lε
Mε is an order-reversing bijection pairing.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 8 / 27
Galois connexions
DefinitionA Galois connexion between two complete lattices L and M is a pair of maps
L
M,
satisfying:x1 ≤ x2 ⇒ x∗2 ≤ x∗1 and y1 ≤ y2 ⇒ y2 ≤ y1 , for all x1, x2 ∈ L, y1, y2 ∈ M.x ≤ x∗ and y ≤ y∗, for all x ∈ L, y ∈ M.
(∨xi)∗ = ∧x∗i
and (∨yi) = ∧yi
, for all subsets xi ⊆ L, yi ⊆ M.(0L)∗ = 1M and (0M) = 1L.
L
M Galois connexion.
The maps ε : x → x∗ and ε : y → y∗ are closure operations in L and M.
Lε = ε-closed ideals of L) and Mε = ε-closed ideals of M are complete
lattice, and Lε
Mε is an order-reversing bijection pairing.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 8 / 27
Galois connexions
DefinitionA Galois connexion between two complete lattices L and M is a pair of maps
L
M,
satisfying:x1 ≤ x2 ⇒ x∗2 ≤ x∗1 and y1 ≤ y2 ⇒ y2 ≤ y1 , for all x1, x2 ∈ L, y1, y2 ∈ M.x ≤ x∗ and y ≤ y∗, for all x ∈ L, y ∈ M.
(∨xi)∗ = ∧x∗i
and (∨yi) = ∧yi
, for all subsets xi ⊆ L, yi ⊆ M.
(0L)∗ = 1M and (0M) = 1L.
L
M Galois connexion.
The maps ε : x → x∗ and ε : y → y∗ are closure operations in L and M.
Lε = ε-closed ideals of L) and Mε = ε-closed ideals of M are complete
lattice, and Lε
Mε is an order-reversing bijection pairing.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 8 / 27
Galois connexions
DefinitionA Galois connexion between two complete lattices L and M is a pair of maps
L
M,
satisfying:x1 ≤ x2 ⇒ x∗2 ≤ x∗1 and y1 ≤ y2 ⇒ y2 ≤ y1 , for all x1, x2 ∈ L, y1, y2 ∈ M.x ≤ x∗ and y ≤ y∗, for all x ∈ L, y ∈ M.
(∨xi)∗ = ∧x∗i
and (∨yi) = ∧yi
, for all subsets xi ⊆ L, yi ⊆ M.
(0L)∗ = 1M and (0M) = 1L.
L
M Galois connexion.
The maps ε : x → x∗ and ε : y → y∗ are closure operations in L and M.
Lε = ε-closed ideals of L) and Mε = ε-closed ideals of M are complete
lattice, and Lε
Mε is an order-reversing bijection pairing.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 8 / 27
Galois connexions
DefinitionA Galois connexion between two complete lattices L and M is a pair of maps
L
M,
satisfying:x1 ≤ x2 ⇒ x∗2 ≤ x∗1 and y1 ≤ y2 ⇒ y2 ≤ y1 , for all x1, x2 ∈ L, y1, y2 ∈ M.x ≤ x∗ and y ≤ y∗, for all x ∈ L, y ∈ M.(∨xi)∗ = ∧x∗
iand (∨yi) = ∧y
i, for all subsets xi ⊆ L, yi ⊆ M.
(0L)∗ = 1M and (0M) = 1L.
L
M Galois connexion.
The maps ε : x → x∗ and ε : y → y∗ are closure operations in L and M.
Lε = ε-closed ideals of L) and Mε = ε-closed ideals of M are complete
lattice, and Lε
Mε is an order-reversing bijection pairing.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 8 / 27
Galois connexions
DefinitionA Galois connexion between two complete lattices L and M is a pair of maps
L
M,
satisfying:x1 ≤ x2 ⇒ x∗2 ≤ x∗1 and y1 ≤ y2 ⇒ y2 ≤ y1 , for all x1, x2 ∈ L, y1, y2 ∈ M.x ≤ x∗ and y ≤ y∗, for all x ∈ L, y ∈ M.(∨xi)∗ = ∧x∗
iand (∨yi) = ∧y
i, for all subsets xi ⊆ L, yi ⊆ M.
(0L)∗ = 1M and (0M) = 1L.
L
M Galois connexion.
The maps ε : x → x∗ and ε : y → y∗ are closure operations in L and M.
Lε = ε-closed ideals of L) and Mε = ε-closed ideals of M are complete
lattice, and Lε
Mε is an order-reversing bijection pairing.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 8 / 27
Galois connexions
DefinitionA Galois connexion between two complete lattices L and M is a pair of maps
L
M,
satisfying:x1 ≤ x2 ⇒ x∗2 ≤ x∗1 and y1 ≤ y2 ⇒ y2 ≤ y1 , for all x1, x2 ∈ L, y1, y2 ∈ M.x ≤ x∗ and y ≤ y∗, for all x ∈ L, y ∈ M.(∨xi)∗ = ∧x∗
iand (∨yi) = ∧y
i, for all subsets xi ⊆ L, yi ⊆ M.
(0L)∗ = 1M and (0M) = 1L.
L
M Galois connexion.
The maps ε : x → x∗ and ε : y → y∗ are closure operations in L and M.
Lε = ε-closed ideals of L) and Mε = ε-closed ideals of M are complete
lattice, and Lε
Mε is an order-reversing bijection pairing.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 8 / 27
Galois connexions
DefinitionA Galois connexion between two complete lattices L and M is a pair of maps
L
M,
satisfying:x1 ≤ x2 ⇒ x∗2 ≤ x∗1 and y1 ≤ y2 ⇒ y2 ≤ y1 , for all x1, x2 ∈ L, y1, y2 ∈ M.x ≤ x∗ and y ≤ y∗, for all x ∈ L, y ∈ M.(∨xi)∗ = ∧x∗
iand (∨yi) = ∧y
i, for all subsets xi ⊆ L, yi ⊆ M.
(0L)∗ = 1M and (0M) = 1L.
L
M Galois connexion.
The maps ε : x → x∗ and ε : y → y∗ are closure operations in L and M.
Lε = ε-closed ideals of L) and Mε = ε-closed ideals of M are complete
lattice, and Lε
Mε is an order-reversing bijection pairing.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 8 / 27
Galois connexions
DefinitionA Galois connexion between two complete lattices L and M is a pair of maps
L
M,
satisfying:x1 ≤ x2 ⇒ x∗2 ≤ x∗1 and y1 ≤ y2 ⇒ y2 ≤ y1 , for all x1, x2 ∈ L, y1, y2 ∈ M.x ≤ x∗ and y ≤ y∗, for all x ∈ L, y ∈ M.(∨xi)∗ = ∧x∗
iand (∨yi) = ∧y
i, for all subsets xi ⊆ L, yi ⊆ M.
(0L)∗ = 1M and (0M) = 1L.
L
M Galois connexion.
The maps ε : x → x∗ and ε : y → y∗ are closure operations in L and M.
Lε = ε-closed ideals of L) and Mε = ε-closed ideals of M are complete
lattice, and Lε
Mε is an order-reversing bijection pairing.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 8 / 27
Galois connexions
DefinitionA Galois connexion between two complete lattices L and M is a pair of maps
L
M,
satisfying:x1 ≤ x2 ⇒ x∗2 ≤ x∗1 and y1 ≤ y2 ⇒ y2 ≤ y1 , for all x1, x2 ∈ L, y1, y2 ∈ M.x ≤ x∗ and y ≤ y∗, for all x ∈ L, y ∈ M.(∨xi)∗ = ∧x∗
iand (∨yi) = ∧y
i, for all subsets xi ⊆ L, yi ⊆ M.
(0L)∗ = 1M and (0M) = 1L.
L
M Galois connexion.
The maps ε : x → x∗ and ε : y → y∗ are closure operations in L and M.
Lε = ε-closed ideals of L) and Mε = ε-closed ideals of M are complete
lattice, and Lε
Mε is an order-reversing bijection pairing.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 8 / 27
Galois connexions
DefinitionA Galois connexion between two complete lattices L and M is a pair of maps
L
M,
satisfying:x1 ≤ x2 ⇒ x∗2 ≤ x∗1 and y1 ≤ y2 ⇒ y2 ≤ y1 , for all x1, x2 ∈ L, y1, y2 ∈ M.x ≤ x∗ and y ≤ y∗, for all x ∈ L, y ∈ M.(∨xi)∗ = ∧x∗
iand (∨yi) = ∧y
i, for all subsets xi ⊆ L, yi ⊆ M.
(0L)∗ = 1M and (0M) = 1L.
L
M Galois connexion.
The maps ε : x → x∗ and ε : y → y∗ are closure operations in L and M.
Lε = ε-closed ideals of L) and Mε = ε-closed ideals of M are complete
lattice, and Lε
Mε is an order-reversing bijection pairing.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 8 / 27
Example
H Hilbert space,
the pairing SH
⊥⊥
SH , where
S⊥ := x ∈ H : < x , S >= 0, ∀S ∈ SH
is a Galois connexion. The ||.||-closure is the closure operation determined bythis Galois connexion:
S||.||
= (S⊥)⊥ ∀S ∈ SH
The map S → S⊥ is an order-reversing bijection from SH
||.|| onto itself.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 9 / 27
Example
H Hilbert space, the pairing SH
⊥⊥
SH , where
S⊥ := x ∈ H : < x , S >= 0, ∀S ∈ SH
is a Galois connexion. The ||.||-closure is the closure operation determined bythis Galois connexion:
S||.||
= (S⊥)⊥ ∀S ∈ SH
The map S → S⊥ is an order-reversing bijection from SH
||.|| onto itself.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 9 / 27
Example
H Hilbert space, the pairing SH
⊥⊥
SH , where
S⊥ := x ∈ H : < x , S >= 0, ∀S ∈ SH
is a Galois connexion.
The ||.||-closure is the closure operation determined bythis Galois connexion:
S||.||
= (S⊥)⊥ ∀S ∈ SH
The map S → S⊥ is an order-reversing bijection from SH
||.|| onto itself.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 9 / 27
Example
H Hilbert space, the pairing SH
⊥⊥
SH , where
S⊥ := x ∈ H : < x , S >= 0, ∀S ∈ SH
is a Galois connexion. The ||.||-closure is the closure operation determined bythis Galois connexion:
S||.||
= (S⊥)⊥ ∀S ∈ SH
The map S → S⊥ is an order-reversing bijection from SH
||.|| onto itself.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 9 / 27
Example
H Hilbert space, the pairing SH
⊥⊥
SH , where
S⊥ := x ∈ H : < x , S >= 0, ∀S ∈ SH
is a Galois connexion. The ||.||-closure is the closure operation determined bythis Galois connexion:
S||.||
= (S⊥)⊥ ∀S ∈ SH
The map S → S⊥ is an order-reversing bijection from SH
||.|| onto itself.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 9 / 27
Example
H Hilbert space, the pairing SH
⊥⊥
SH , where
S⊥ := x ∈ H : < x , S >= 0, ∀S ∈ SH
is a Galois connexion. The ||.||-closure is the closure operation determined bythis Galois connexion:
S||.||
= (S⊥)⊥ ∀S ∈ SH
The map S → S⊥ is an order-reversing bijection from SH
||.|| onto itself.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 9 / 27
Example
X normed space and X ∗ its topological dual space.
The pairing SX
(.)0
(.)0
SX∗ defined by
S → S0 := x∗ ∈ X ∗ : x∗(S) = 0 and V → V0 := x ∈ X : V (x) = 0is a Galois connexion and the closure operations in SX and SX∗ are
ε(S) = (S0)0 = Sσ(X ,X∗)
and ε(V ) = (V0)0 = Vσ(X∗,X)
.
Sσ(X ,X∗)X
and Sσ(X∗,X)X∗ are complete lattice, and
Sσ(X ,X∗)X
(.)0
(.)0
Sσ(X∗,X)X∗
is an order-reversing bijective pairing.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 10 / 27
ExampleX normed space and X ∗ its topological dual space.
The pairing SX
(.)0
(.)0
SX∗ defined by
S → S0 := x∗ ∈ X ∗ : x∗(S) = 0 and V → V0 := x ∈ X : V (x) = 0is a Galois connexion and the closure operations in SX and SX∗ are
ε(S) = (S0)0 = Sσ(X ,X∗)
and ε(V ) = (V0)0 = Vσ(X∗,X)
.
Sσ(X ,X∗)X
and Sσ(X∗,X)X∗ are complete lattice, and
Sσ(X ,X∗)X
(.)0
(.)0
Sσ(X∗,X)X∗
is an order-reversing bijective pairing.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 10 / 27
ExampleX normed space and X ∗ its topological dual space.
The pairing SX
(.)0
(.)0
SX∗ defined by
S → S0 := x∗ ∈ X ∗ : x∗(S) = 0 and V → V0 := x ∈ X : V (x) = 0is a Galois connexion
and the closure operations in SX and SX∗ areε(S) = (S0)0 = S
σ(X ,X∗)and ε(V ) = (V0)0 = V
σ(X∗,X).
Sσ(X ,X∗)X
and Sσ(X∗,X)X∗ are complete lattice, and
Sσ(X ,X∗)X
(.)0
(.)0
Sσ(X∗,X)X∗
is an order-reversing bijective pairing.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 10 / 27
ExampleX normed space and X ∗ its topological dual space.
The pairing SX
(.)0
(.)0
SX∗ defined by
S → S0 := x∗ ∈ X ∗ : x∗(S) = 0 and V → V0 := x ∈ X : V (x) = 0is a Galois connexion and the closure operations in SX and SX∗ are
ε(S) = (S0)0 = Sσ(X ,X∗)
and ε(V ) = (V0)0 = Vσ(X∗,X)
.
Sσ(X ,X∗)X
and Sσ(X∗,X)X∗ are complete lattice, and
Sσ(X ,X∗)X
(.)0
(.)0
Sσ(X∗,X)X∗
is an order-reversing bijective pairing.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 10 / 27
ExampleX normed space and X ∗ its topological dual space.
The pairing SX
(.)0
(.)0
SX∗ defined by
S → S0 := x∗ ∈ X ∗ : x∗(S) = 0 and V → V0 := x ∈ X : V (x) = 0is a Galois connexion and the closure operations in SX and SX∗ are
ε(S) = (S0)0 = Sσ(X ,X∗)
and ε(V ) = (V0)0 = Vσ(X∗,X)
.
Sσ(X ,X∗)X
and Sσ(X∗,X)X∗ are complete lattice, and
Sσ(X ,X∗)X
(.)0
(.)0
Sσ(X∗,X)X∗
is an order-reversing bijective pairing.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 10 / 27
ExampleX normed space and X ∗ its topological dual space.
The pairing SX
(.)0
(.)0
SX∗ defined by
S → S0 := x∗ ∈ X ∗ : x∗(S) = 0 and V → V0 := x ∈ X : V (x) = 0is a Galois connexion and the closure operations in SX and SX∗ are
ε(S) = (S0)0 = Sσ(X ,X∗)
and ε(V ) = (V0)0 = Vσ(X∗,X)
.
Sσ(X ,X∗)X
and Sσ(X∗,X)X∗ are complete lattice, and
Sσ(X ,X∗)X
(.)0
(.)0
Sσ(X∗,X)X∗
is an order-reversing bijective pairing.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 10 / 27
algebraic closures: π-closure and ε-closure
I ideal of an algebra A, the annihilator of I in A, Ann(I), is the largest ideal J
of A satisfying the conditions IJ = JI = 0.
Definition
The pairing IA
Ann(.)
Ann(.)IA is a Galois connexion.
The π-closure is the closure associated to this Galois connexion, that is,I = Ann(Ann(I)).
IπA
= π-closed ideals of A is a complete lattice.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 11 / 27
algebraic closures: π-closure and ε-closure
I ideal of an algebra A,
the annihilator of I in A, Ann(I), is the largest ideal J
of A satisfying the conditions IJ = JI = 0.
Definition
The pairing IA
Ann(.)
Ann(.)IA is a Galois connexion.
The π-closure is the closure associated to this Galois connexion, that is,I = Ann(Ann(I)).
IπA
= π-closed ideals of A is a complete lattice.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 11 / 27
algebraic closures: π-closure and ε-closure
I ideal of an algebra A, the annihilator of I in A, Ann(I), is the largest ideal J
of A satisfying the conditions IJ = JI = 0.
Definition
The pairing IA
Ann(.)
Ann(.)IA is a Galois connexion.
The π-closure is the closure associated to this Galois connexion, that is,I = Ann(Ann(I)).
IπA
= π-closed ideals of A is a complete lattice.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 11 / 27
algebraic closures: π-closure and ε-closure
I ideal of an algebra A, the annihilator of I in A, Ann(I), is the largest ideal J
of A satisfying the conditions IJ = JI = 0.
Definition
The pairing IA
Ann(.)
Ann(.)IA is a Galois connexion.
The π-closure is the closure associated to this Galois connexion, that is,I = Ann(Ann(I)).
IπA
= π-closed ideals of A is a complete lattice.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 11 / 27
algebraic closures: π-closure and ε-closure
I ideal of an algebra A, the annihilator of I in A, Ann(I), is the largest ideal J
of A satisfying the conditions IJ = JI = 0.
Definition
The pairing IA
Ann(.)
Ann(.)IA is a Galois connexion.
The π-closure is the closure associated to this Galois connexion, that is,I = Ann(Ann(I)).
IπA
= π-closed ideals of A is a complete lattice.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 11 / 27
algebraic closures: π-closure and ε-closure
I ideal of an algebra A, the annihilator of I in A, Ann(I), is the largest ideal J
of A satisfying the conditions IJ = JI = 0.
Definition
The pairing IA
Ann(.)
Ann(.)IA is a Galois connexion.
The π-closure is the closure associated to this Galois connexion, that is,
I = Ann(Ann(I)).
IπA
= π-closed ideals of A is a complete lattice.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 11 / 27
algebraic closures: π-closure and ε-closure
I ideal of an algebra A, the annihilator of I in A, Ann(I), is the largest ideal J
of A satisfying the conditions IJ = JI = 0.
Definition
The pairing IA
Ann(.)
Ann(.)IA is a Galois connexion.
The π-closure is the closure associated to this Galois connexion, that is,I = Ann(Ann(I)).
IπA
= π-closed ideals of A is a complete lattice.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 11 / 27
algebraic closures: π-closure and ε-closure
I ideal of an algebra A, the annihilator of I in A, Ann(I), is the largest ideal J
of A satisfying the conditions IJ = JI = 0.
Definition
The pairing IA
Ann(.)
Ann(.)IA is a Galois connexion.
The π-closure is the closure associated to this Galois connexion, that is,I = Ann(Ann(I)).
IπA
= π-closed ideals of A is a complete lattice.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 11 / 27
algebraic closures: π-closure and ε-closure
I ideal of an algebra A, the annihilator of I in A, Ann(I), is the largest ideal J
of A satisfying the conditions IJ = JI = 0.
Definition
The pairing IA
Ann(.)
Ann(.)IA is a Galois connexion.
The π-closure is the closure associated to this Galois connexion, that is,I = Ann(Ann(I)).
IπA
= π-closed ideals of A is a complete lattice.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 11 / 27
algebraic closures:
π-closure and ε-closure
I ideal of an algebra A, the annihilator of I in A, Ann(I), is the largest ideal J
of A satisfying the conditions IJ = JI = 0.
Definition
The pairing IA
Ann(.)
Ann(.)IA is a Galois connexion.
The π-closure is the closure associated to this Galois connexion, that is,I = Ann(Ann(I)).
IπA
= π-closed ideals of A is a complete lattice.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 11 / 27
algebraic closures: π-closure and ε-closure
I ideal of an algebra A, the annihilator of I in A, Ann(I), is the largest ideal J
of A satisfying the conditions IJ = JI = 0.
Definition
The pairing IA
Ann(.)
Ann(.)IA is a Galois connexion.
The π-closure is the closure associated to this Galois connexion, that is,I = Ann(Ann(I)).
IπA
= π-closed ideals of A is a complete lattice.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 11 / 27
The π-closure determine some algebraic properties.
DefinitionAn algebra A is said to be prime if, for ideals I and J of A, the condition IJ = 0implies either I = 0 or J = 0.
PropositionA algebra, A2 = 0. (non null)A is prime ⇐⇒ Iπ
A= 0, A.
DefinitionA is semiprime if 0 is the unique ideal I of A with I2 = 0.
PropositionA semiprime ⇐⇒ A = I ⊕ Ann(I) (∀I ∈ IA)
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 12 / 27
The π-closure determine some algebraic properties.
DefinitionAn algebra A is said to be prime if, for ideals I and J of A, the condition IJ = 0implies either I = 0 or J = 0.
PropositionA algebra, A2 = 0. (non null)A is prime ⇐⇒ Iπ
A= 0, A.
DefinitionA is semiprime if 0 is the unique ideal I of A with I2 = 0.
PropositionA semiprime ⇐⇒ A = I ⊕ Ann(I) (∀I ∈ IA)
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 12 / 27
The π-closure determine some algebraic properties.
DefinitionAn algebra A is said to be prime if, for ideals I and J of A, the condition IJ = 0implies either I = 0 or J = 0.
Proposition
A algebra, A2 = 0. (non null)A is prime ⇐⇒ Iπ
A= 0, A.
DefinitionA is semiprime if 0 is the unique ideal I of A with I2 = 0.
PropositionA semiprime ⇐⇒ A = I ⊕ Ann(I) (∀I ∈ IA)
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 12 / 27
The π-closure determine some algebraic properties.
DefinitionAn algebra A is said to be prime if, for ideals I and J of A, the condition IJ = 0implies either I = 0 or J = 0.
PropositionA algebra, A2 = 0. (non null)
A is prime ⇐⇒ IπA
= 0, A.
DefinitionA is semiprime if 0 is the unique ideal I of A with I2 = 0.
PropositionA semiprime ⇐⇒ A = I ⊕ Ann(I) (∀I ∈ IA)
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 12 / 27
The π-closure determine some algebraic properties.
DefinitionAn algebra A is said to be prime if, for ideals I and J of A, the condition IJ = 0implies either I = 0 or J = 0.
PropositionA algebra, A2 = 0. (non null)A is prime ⇐⇒ Iπ
A= 0, A.
DefinitionA is semiprime if 0 is the unique ideal I of A with I2 = 0.
PropositionA semiprime ⇐⇒ A = I ⊕ Ann(I) (∀I ∈ IA)
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 12 / 27
The π-closure determine some algebraic properties.
DefinitionAn algebra A is said to be prime if, for ideals I and J of A, the condition IJ = 0implies either I = 0 or J = 0.
PropositionA algebra, A2 = 0. (non null)A is prime ⇐⇒ Iπ
A= 0, A.
DefinitionA is semiprime if 0 is the unique ideal I of A with I2 = 0.
PropositionA semiprime ⇐⇒ A = I ⊕ Ann(I) (∀I ∈ IA)
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 12 / 27
The π-closure determine some algebraic properties.
DefinitionAn algebra A is said to be prime if, for ideals I and J of A, the condition IJ = 0implies either I = 0 or J = 0.
PropositionA algebra, A2 = 0. (non null)A is prime ⇐⇒ Iπ
A= 0, A.
DefinitionA is semiprime if 0 is the unique ideal I of A with I2 = 0.
Proposition
A semiprime ⇐⇒ A = I ⊕ Ann(I) (∀I ∈ IA)
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 12 / 27
The π-closure determine some algebraic properties.
DefinitionAn algebra A is said to be prime if, for ideals I and J of A, the condition IJ = 0implies either I = 0 or J = 0.
PropositionA algebra, A2 = 0. (non null)A is prime ⇐⇒ Iπ
A= 0, A.
DefinitionA is semiprime if 0 is the unique ideal I of A with I2 = 0.
PropositionA semiprime ⇐⇒ A = I ⊕ Ann(I) (∀I ∈ IA)
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 12 / 27
The π-closure determine some algebraic properties.
DefinitionAn algebra A is said to be prime if, for ideals I and J of A, the condition IJ = 0implies either I = 0 or J = 0.
PropositionA algebra, A2 = 0. (non null)A is prime ⇐⇒ Iπ
A= 0, A.
DefinitionA is semiprime if 0 is the unique ideal I of A with I2 = 0.
PropositionA semiprime ⇐⇒ A = I ⊕ Ann(I) (∀I ∈ IA)
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 12 / 27
ε-closure
A algebra, S subspace of A.
Sann = F ∈ M(A) : F (x) = 0 for each x ∈ S.
N subspace of M(A),
Nann = a ∈ A : F (a) = 0 for each F ∈ N.
DefinitionA algebra. The pairing
IA
(.)ann
(.)ann
IM(A)
is a Galois connexion. The ε-closure is the closure in IA associated to thisGalois connexion,
I∧ = (Iann)ann I ∈ IA.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 13 / 27
ε-closure
A algebra, S subspace of A.
Sann = F ∈ M(A) : F (x) = 0 for each x ∈ S.
N subspace of M(A),
Nann = a ∈ A : F (a) = 0 for each F ∈ N.
DefinitionA algebra. The pairing
IA
(.)ann
(.)ann
IM(A)
is a Galois connexion. The ε-closure is the closure in IA associated to thisGalois connexion,
I∧ = (Iann)ann I ∈ IA.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 13 / 27
ε-closure
A algebra, S subspace of A.
Sann = F ∈ M(A) : F (x) = 0 for each x ∈ S.
N subspace of M(A),
Nann = a ∈ A : F (a) = 0 for each F ∈ N.
DefinitionA algebra. The pairing
IA
(.)ann
(.)ann
IM(A)
is a Galois connexion. The ε-closure is the closure in IA associated to thisGalois connexion,
I∧ = (Iann)ann I ∈ IA.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 13 / 27
ε-closure
A algebra, S subspace of A.
Sann = F ∈ M(A) : F (x) = 0 for each x ∈ S.
N subspace of M(A),
Nann = a ∈ A : F (a) = 0 for each F ∈ N.
DefinitionA algebra. The pairing
IA
(.)ann
(.)ann
IM(A)
is a Galois connexion. The ε-closure is the closure in IA associated to thisGalois connexion,
I∧ = (Iann)ann I ∈ IA.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 13 / 27
ε-closure
A algebra, S subspace of A.
Sann = F ∈ M(A) : F (x) = 0 for each x ∈ S.
N subspace of M(A),
Nann = a ∈ A : F (a) = 0 for each F ∈ N.
DefinitionA algebra. The pairing
IA
(.)ann
(.)ann
IM(A)
is a Galois connexion. The ε-closure is the closure in IA associated to thisGalois connexion,
I∧ = (Iann)ann I ∈ IA.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 13 / 27
ε-closure
A algebra, S subspace of A.
Sann = F ∈ M(A) : F (x) = 0 for each x ∈ S.
N subspace of M(A),
Nann = a ∈ A : F (a) = 0 for each F ∈ N.
Definition
A algebra. The pairing
IA
(.)ann
(.)ann
IM(A)
is a Galois connexion. The ε-closure is the closure in IA associated to thisGalois connexion,
I∧ = (Iann)ann I ∈ IA.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 13 / 27
ε-closure
A algebra, S subspace of A.
Sann = F ∈ M(A) : F (x) = 0 for each x ∈ S.
N subspace of M(A),
Nann = a ∈ A : F (a) = 0 for each F ∈ N.
DefinitionA algebra.
The pairing
IA
(.)ann
(.)ann
IM(A)
is a Galois connexion. The ε-closure is the closure in IA associated to thisGalois connexion,
I∧ = (Iann)ann I ∈ IA.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 13 / 27
ε-closure
A algebra, S subspace of A.
Sann = F ∈ M(A) : F (x) = 0 for each x ∈ S.
N subspace of M(A),
Nann = a ∈ A : F (a) = 0 for each F ∈ N.
DefinitionA algebra. The pairing
IA
(.)ann
(.)ann
IM(A)
is a Galois connexion. The ε-closure is the closure in IA associated to thisGalois connexion,
I∧ = (Iann)ann I ∈ IA.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 13 / 27
ε-closure
A algebra, S subspace of A.
Sann = F ∈ M(A) : F (x) = 0 for each x ∈ S.
N subspace of M(A),
Nann = a ∈ A : F (a) = 0 for each F ∈ N.
DefinitionA algebra. The pairing
IA
(.)ann
(.)ann
IM(A)
is a Galois connexion.
The ε-closure is the closure in IA associated to thisGalois connexion,
I∧ = (Iann)ann I ∈ IA.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 13 / 27
ε-closure
A algebra, S subspace of A.
Sann = F ∈ M(A) : F (x) = 0 for each x ∈ S.
N subspace of M(A),
Nann = a ∈ A : F (a) = 0 for each F ∈ N.
DefinitionA algebra. The pairing
IA
(.)ann
(.)ann
IM(A)
is a Galois connexion. The ε-closure is the closure in IA associated to thisGalois connexion,
I∧ = (Iann)ann I ∈ IA.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 13 / 27
ε-closure
A algebra, S subspace of A.
Sann = F ∈ M(A) : F (x) = 0 for each x ∈ S.
N subspace of M(A),
Nann = a ∈ A : F (a) = 0 for each F ∈ N.
DefinitionA algebra. The pairing
IA
(.)ann
(.)ann
IM(A)
is a Galois connexion. The ε-closure is the closure in IA associated to thisGalois connexion,
I∧ = (Iann)ann I ∈ IA.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 13 / 27
Properties of the ε-closure
PropositionM(A) semiprime =⇒ A = (I + Ann(I))∧ (∀I ∈ IA)
Continuity property
F ∈ M(A), I ∈ IA =⇒ F (I ∧) ⊆ F (I)∧.
Relationships between closuresA algebra, I ideal de A.
I ⊆ I;Ann(I)∧ = Ann(I ∧) = Ann(I);If A is normed (endowed with a norm ||.|| satisfying ||ab|| ≤ ||a||||b||, forall a, b ∈ A) then
I ⊆ I||.|| ⊆ I ⊆ I.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 14 / 27
Properties of the ε-closure
PropositionM(A) semiprime =⇒ A = (I + Ann(I))∧ (∀I ∈ IA)
Continuity property
F ∈ M(A), I ∈ IA =⇒ F (I ∧) ⊆ F (I)∧.
Relationships between closuresA algebra, I ideal de A.
I ⊆ I;Ann(I)∧ = Ann(I ∧) = Ann(I);If A is normed (endowed with a norm ||.|| satisfying ||ab|| ≤ ||a||||b||, forall a, b ∈ A) then
I ⊆ I||.|| ⊆ I ⊆ I.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 14 / 27
Properties of the ε-closure
PropositionM(A) semiprime =⇒ A = (I + Ann(I))∧ (∀I ∈ IA)
Continuity property
F ∈ M(A), I ∈ IA =⇒ F (I ∧) ⊆ F (I)∧.
Relationships between closuresA algebra, I ideal de A.
I ⊆ I;Ann(I)∧ = Ann(I ∧) = Ann(I);If A is normed (endowed with a norm ||.|| satisfying ||ab|| ≤ ||a||||b||, forall a, b ∈ A) then
I ⊆ I||.|| ⊆ I ⊆ I.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 14 / 27
Properties of the ε-closure
PropositionM(A) semiprime =⇒ A = (I + Ann(I))∧ (∀I ∈ IA)
Continuity property
F ∈ M(A), I ∈ IA =⇒ F (I ∧) ⊆ F (I)∧.
Relationships between closuresA algebra, I ideal de A.
I ⊆ I;Ann(I)∧ = Ann(I ∧) = Ann(I);If A is normed (endowed with a norm ||.|| satisfying ||ab|| ≤ ||a||||b||, forall a, b ∈ A) then
I ⊆ I||.|| ⊆ I ⊆ I.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 14 / 27
Properties of the ε-closure
PropositionM(A) semiprime =⇒ A = (I + Ann(I))∧ (∀I ∈ IA)
Continuity property
F ∈ M(A), I ∈ IA =⇒ F (I ∧) ⊆ F (I)∧.
Relationships between closuresA algebra, I ideal de A.
I ⊆ I;Ann(I)∧ = Ann(I ∧) = Ann(I);If A is normed (endowed with a norm ||.|| satisfying ||ab|| ≤ ||a||||b||, forall a, b ∈ A) then
I ⊆ I||.|| ⊆ I ⊆ I.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 14 / 27
Properties of the ε-closure
PropositionM(A) semiprime =⇒ A = (I + Ann(I))∧ (∀I ∈ IA)
Continuity property
F ∈ M(A), I ∈ IA =⇒ F (I ∧) ⊆ F (I)∧.
Relationships between closuresA algebra, I ideal de A.
I ⊆ I;Ann(I)∧ = Ann(I ∧) = Ann(I);If A is normed (endowed with a norm ||.|| satisfying ||ab|| ≤ ||a||||b||, forall a, b ∈ A) then
I ⊆ I||.|| ⊆ I ⊆ I.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 14 / 27
Properties of the ε-closure
PropositionM(A) semiprime =⇒ A = (I + Ann(I))∧ (∀I ∈ IA)
Continuity property
F ∈ M(A), I ∈ IA =⇒ F (I ∧) ⊆ F (I)∧.
Relationships between closuresA algebra, I ideal de A.
I ⊆ I;Ann(I)∧ = Ann(I ∧) = Ann(I);If A is normed (endowed with a norm ||.|| satisfying ||ab|| ≤ ||a||||b||, forall a, b ∈ A) then
I ⊆ I||.|| ⊆ I ⊆ I.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 14 / 27
Properties of the ε-closure
PropositionM(A) semiprime =⇒ A = (I + Ann(I))∧ (∀I ∈ IA)
Continuity property
F ∈ M(A), I ∈ IA =⇒ F (I ∧) ⊆ F (I)∧.
Relationships between closuresA algebra, I ideal de A.
I ⊆ I;
Ann(I)∧ = Ann(I ∧) = Ann(I);
If A is normed (endowed with a norm ||.|| satisfying ||ab|| ≤ ||a||||b||, forall a, b ∈ A) then
I ⊆ I||.|| ⊆ I ⊆ I.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 14 / 27
Properties of the ε-closure
PropositionM(A) semiprime =⇒ A = (I + Ann(I))∧ (∀I ∈ IA)
Continuity property
F ∈ M(A), I ∈ IA =⇒ F (I ∧) ⊆ F (I)∧.
Relationships between closuresA algebra, I ideal de A.
I ⊆ I;Ann(I)∧ = Ann(I ∧) = Ann(I);
If A is normed (endowed with a norm ||.|| satisfying ||ab|| ≤ ||a||||b||, forall a, b ∈ A) then
I ⊆ I||.|| ⊆ I ⊆ I.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 14 / 27
Properties of the ε-closure
PropositionM(A) semiprime =⇒ A = (I + Ann(I))∧ (∀I ∈ IA)
Continuity property
F ∈ M(A), I ∈ IA =⇒ F (I ∧) ⊆ F (I)∧.
Relationships between closuresA algebra, I ideal de A.
I ⊆ I;Ann(I)∧ = Ann(I ∧) = Ann(I);
If A is normed (endowed with a norm ||.|| satisfying ||ab|| ≤ ||a||||b||, forall a, b ∈ A) then
I ⊆ I||.|| ⊆ I ⊆ I.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 14 / 27
Properties of the ε-closure
PropositionM(A) semiprime =⇒ A = (I + Ann(I))∧ (∀I ∈ IA)
Continuity property
F ∈ M(A), I ∈ IA =⇒ F (I ∧) ⊆ F (I)∧.
Relationships between closuresA algebra, I ideal de A.
I ⊆ I;Ann(I)∧ = Ann(I ∧) = Ann(I);If A is normed (endowed with a norm ||.|| satisfying ||ab|| ≤ ||a||||b||, forall a, b ∈ A) then
I ⊆ I||.|| ⊆ I ⊆ I.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 14 / 27
Properties of the ε-closure
PropositionM(A) semiprime =⇒ A = (I + Ann(I))∧ (∀I ∈ IA)
Continuity property
F ∈ M(A), I ∈ IA =⇒ F (I ∧) ⊆ F (I)∧.
Relationships between closuresA algebra, I ideal de A.
I ⊆ I;Ann(I)∧ = Ann(I ∧) = Ann(I);If A is normed (endowed with a norm ||.|| satisfying ||ab|| ≤ ||a||||b||, forall a, b ∈ A) then
I ⊆ I||.|| ⊆ I ⊆ I.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 14 / 27
Properties of the ε-closure
PropositionM(A) semiprime =⇒ A = (I + Ann(I))∧ (∀I ∈ IA)
Continuity property
F ∈ M(A), I ∈ IA =⇒ F (I ∧) ⊆ F (I)∧.
Relationships between closuresA algebra, I ideal de A.
I ⊆ I;Ann(I)∧ = Ann(I ∧) = Ann(I);If A is normed (endowed with a norm ||.|| satisfying ||ab|| ≤ ||a||||b||, forall a, b ∈ A) then
I ⊆ I||.|| ⊆ I ⊆ I.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 14 / 27
Relationships between the semiprimeness of A andM(A)
A semiprime ⇒ M(A) semiprime
Example: Algebra of Albert (three-dimensional)The algebra generated by 1, u, v, whose product is defined by:
u2 = 1, uv = v2 = v , vu = 0.
M(A) semiprime ⇒ A semiprimeExample: A quotient algebra of the associative free algebra
DefinitionAn algebra A is multiplicatively semiprime ( m.s.p.) whenever both A andM(A) are semiprime algebras.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 15 / 27
Relationships between the semiprimeness of A andM(A)
A semiprime ⇒ M(A) semiprime
Example: Algebra of Albert (three-dimensional)The algebra generated by 1, u, v, whose product is defined by:
u2 = 1, uv = v2 = v , vu = 0.
M(A) semiprime ⇒ A semiprimeExample: A quotient algebra of the associative free algebra
DefinitionAn algebra A is multiplicatively semiprime ( m.s.p.) whenever both A andM(A) are semiprime algebras.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 15 / 27
Relationships between the semiprimeness of A andM(A)
A semiprime ⇒ M(A) semiprimeExample: Algebra of Albert (three-dimensional)
The algebra generated by 1, u, v, whose product is defined by:u2 = 1, uv = v2 = v , vu = 0.
M(A) semiprime ⇒ A semiprimeExample: A quotient algebra of the associative free algebra
DefinitionAn algebra A is multiplicatively semiprime ( m.s.p.) whenever both A andM(A) are semiprime algebras.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 15 / 27
Relationships between the semiprimeness of A andM(A)
A semiprime ⇒ M(A) semiprimeExample: Algebra of Albert (three-dimensional)The algebra generated by 1, u, v, whose product is defined by:
u2 = 1, uv = v2 = v , vu = 0.
M(A) semiprime ⇒ A semiprimeExample: A quotient algebra of the associative free algebra
DefinitionAn algebra A is multiplicatively semiprime ( m.s.p.) whenever both A andM(A) are semiprime algebras.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 15 / 27
Relationships between the semiprimeness of A andM(A)
A semiprime ⇒ M(A) semiprimeExample: Algebra of Albert (three-dimensional)The algebra generated by 1, u, v, whose product is defined by:
u2 = 1, uv = v2 = v , vu = 0.
M(A) semiprime ⇒ A semiprime
Example: A quotient algebra of the associative free algebra
DefinitionAn algebra A is multiplicatively semiprime ( m.s.p.) whenever both A andM(A) are semiprime algebras.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 15 / 27
Relationships between the semiprimeness of A andM(A)
A semiprime ⇒ M(A) semiprimeExample: Algebra of Albert (three-dimensional)The algebra generated by 1, u, v, whose product is defined by:
u2 = 1, uv = v2 = v , vu = 0.
M(A) semiprime ⇒ A semiprime
Example: A quotient algebra of the associative free algebra
DefinitionAn algebra A is multiplicatively semiprime ( m.s.p.) whenever both A andM(A) are semiprime algebras.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 15 / 27
Relationships between the semiprimeness of A andM(A)
A semiprime ⇒ M(A) semiprimeExample: Algebra of Albert (three-dimensional)The algebra generated by 1, u, v, whose product is defined by:
u2 = 1, uv = v2 = v , vu = 0.
M(A) semiprime ⇒ A semiprimeExample: A quotient algebra of the associative free algebra
DefinitionAn algebra A is multiplicatively semiprime ( m.s.p.) whenever both A andM(A) are semiprime algebras.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 15 / 27
Relationships between the semiprimeness of A andM(A)
A semiprime ⇒ M(A) semiprimeExample: Algebra of Albert (three-dimensional)The algebra generated by 1, u, v, whose product is defined by:
u2 = 1, uv = v2 = v , vu = 0.
M(A) semiprime ⇒ A semiprimeExample: A quotient algebra of the associative free algebra
Definition
An algebra A is multiplicatively semiprime ( m.s.p.) whenever both A andM(A) are semiprime algebras.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 15 / 27
Relationships between the semiprimeness of A andM(A)
A semiprime ⇒ M(A) semiprimeExample: Algebra of Albert (three-dimensional)The algebra generated by 1, u, v, whose product is defined by:
u2 = 1, uv = v2 = v , vu = 0.
M(A) semiprime ⇒ A semiprimeExample: A quotient algebra of the associative free algebra
DefinitionAn algebra A is multiplicatively semiprime ( m.s.p.) whenever both A andM(A) are semiprime algebras.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 15 / 27
The class of multiplicatively semiprime algebras is quite large
Example
nondegenerate alternative algebras (semiprime associative algebras)nondegenerate Jordan algebrasfree nonassociative algebra generated by a nonempty set
TheoremA algebra.A m.s.p. ⇔ A semiprime and ε = π ⇔ A = (I ⊕ Ann(I))∧ (∀I ∈ IA).
TheoremA algebra, I ∈ IA.
A m.s.p. =⇒ I m.s.p.A/I m.s.p.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 16 / 27
The class of multiplicatively semiprime algebras is quite large
Examplenondegenerate alternative algebras
(semiprime associative algebras)
nondegenerate Jordan algebrasfree nonassociative algebra generated by a nonempty set
TheoremA algebra.A m.s.p. ⇔ A semiprime and ε = π ⇔ A = (I ⊕ Ann(I))∧ (∀I ∈ IA).
TheoremA algebra, I ∈ IA.
A m.s.p. =⇒ I m.s.p.A/I m.s.p.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 16 / 27
The class of multiplicatively semiprime algebras is quite large
Examplenondegenerate alternative algebras
(semiprime associative algebras)
nondegenerate Jordan algebrasfree nonassociative algebra generated by a nonempty set
TheoremA algebra.A m.s.p. ⇔ A semiprime and ε = π ⇔ A = (I ⊕ Ann(I))∧ (∀I ∈ IA).
TheoremA algebra, I ∈ IA.
A m.s.p. =⇒ I m.s.p.A/I m.s.p.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 16 / 27
The class of multiplicatively semiprime algebras is quite large
Examplenondegenerate alternative algebras (semiprime associative algebras)
nondegenerate Jordan algebrasfree nonassociative algebra generated by a nonempty set
TheoremA algebra.A m.s.p. ⇔ A semiprime and ε = π ⇔ A = (I ⊕ Ann(I))∧ (∀I ∈ IA).
TheoremA algebra, I ∈ IA.
A m.s.p. =⇒ I m.s.p.A/I m.s.p.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 16 / 27
The class of multiplicatively semiprime algebras is quite large
Examplenondegenerate alternative algebras (semiprime associative algebras)nondegenerate Jordan algebras
free nonassociative algebra generated by a nonempty set
TheoremA algebra.A m.s.p. ⇔ A semiprime and ε = π ⇔ A = (I ⊕ Ann(I))∧ (∀I ∈ IA).
TheoremA algebra, I ∈ IA.
A m.s.p. =⇒ I m.s.p.A/I m.s.p.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 16 / 27
The class of multiplicatively semiprime algebras is quite large
Examplenondegenerate alternative algebras (semiprime associative algebras)nondegenerate Jordan algebrasfree nonassociative algebra generated by a nonempty set
TheoremA algebra.A m.s.p. ⇔ A semiprime and ε = π ⇔ A = (I ⊕ Ann(I))∧ (∀I ∈ IA).
TheoremA algebra, I ∈ IA.
A m.s.p. =⇒ I m.s.p.A/I m.s.p.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 16 / 27
The class of multiplicatively semiprime algebras is quite large
Examplenondegenerate alternative algebras (semiprime associative algebras)nondegenerate Jordan algebrasfree nonassociative algebra generated by a nonempty set
TheoremA algebra.A m.s.p. ⇔ A semiprime and ε = π ⇔ A = (I ⊕ Ann(I))∧ (∀I ∈ IA).
TheoremA algebra, I ∈ IA.
A m.s.p. =⇒ I m.s.p.A/I m.s.p.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 16 / 27
The class of multiplicatively semiprime algebras is quite large
Examplenondegenerate alternative algebras (semiprime associative algebras)nondegenerate Jordan algebrasfree nonassociative algebra generated by a nonempty set
Theorem
A algebra.A m.s.p. ⇔ A semiprime and ε = π ⇔ A = (I ⊕ Ann(I))∧ (∀I ∈ IA).
TheoremA algebra, I ∈ IA.
A m.s.p. =⇒ I m.s.p.A/I m.s.p.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 16 / 27
The class of multiplicatively semiprime algebras is quite large
Examplenondegenerate alternative algebras (semiprime associative algebras)nondegenerate Jordan algebrasfree nonassociative algebra generated by a nonempty set
TheoremA algebra.
A m.s.p. ⇔ A semiprime and ε = π ⇔ A = (I ⊕ Ann(I))∧ (∀I ∈ IA).
TheoremA algebra, I ∈ IA.
A m.s.p. =⇒ I m.s.p.A/I m.s.p.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 16 / 27
The class of multiplicatively semiprime algebras is quite large
Examplenondegenerate alternative algebras (semiprime associative algebras)nondegenerate Jordan algebrasfree nonassociative algebra generated by a nonempty set
TheoremA algebra.A m.s.p. ⇔ A semiprime and ε = π
⇔ A = (I ⊕ Ann(I))∧ (∀I ∈ IA).
TheoremA algebra, I ∈ IA.
A m.s.p. =⇒ I m.s.p.A/I m.s.p.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 16 / 27
The class of multiplicatively semiprime algebras is quite large
Examplenondegenerate alternative algebras (semiprime associative algebras)nondegenerate Jordan algebrasfree nonassociative algebra generated by a nonempty set
TheoremA algebra.A m.s.p. ⇔ A semiprime and ε = π ⇔ A = (I ⊕ Ann(I))∧ (∀I ∈ IA).
TheoremA algebra, I ∈ IA.
A m.s.p. =⇒ I m.s.p.A/I m.s.p.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 16 / 27
The class of multiplicatively semiprime algebras is quite large
Examplenondegenerate alternative algebras (semiprime associative algebras)nondegenerate Jordan algebrasfree nonassociative algebra generated by a nonempty set
TheoremA algebra.A m.s.p. ⇔ A semiprime and ε = π ⇔ A = (I ⊕ Ann(I))∧ (∀I ∈ IA).
Theorem
A algebra, I ∈ IA.
A m.s.p. =⇒ I m.s.p.A/I m.s.p.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 16 / 27
The class of multiplicatively semiprime algebras is quite large
Examplenondegenerate alternative algebras (semiprime associative algebras)nondegenerate Jordan algebrasfree nonassociative algebra generated by a nonempty set
TheoremA algebra.A m.s.p. ⇔ A semiprime and ε = π ⇔ A = (I ⊕ Ann(I))∧ (∀I ∈ IA).
TheoremA algebra, I ∈ IA.
A m.s.p. =⇒ I m.s.p.A/I m.s.p.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 16 / 27
The class of multiplicatively semiprime algebras is quite large
Examplenondegenerate alternative algebras (semiprime associative algebras)nondegenerate Jordan algebrasfree nonassociative algebra generated by a nonempty set
TheoremA algebra.A m.s.p. ⇔ A semiprime and ε = π ⇔ A = (I ⊕ Ann(I))∧ (∀I ∈ IA).
TheoremA algebra, I ∈ IA.
A m.s.p. =⇒ I m.s.p.
A/I m.s.p.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 16 / 27
The class of multiplicatively semiprime algebras is quite large
Examplenondegenerate alternative algebras (semiprime associative algebras)nondegenerate Jordan algebrasfree nonassociative algebra generated by a nonempty set
TheoremA algebra.A m.s.p. ⇔ A semiprime and ε = π ⇔ A = (I ⊕ Ann(I))∧ (∀I ∈ IA).
TheoremA algebra, I ∈ IA.
A m.s.p. =⇒ I m.s.p.A/I m.s.p.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 16 / 27
subspaces complemented
DefinitionX Banach space.A ||.||-closed subspace Y of X is said to be complemented in X if there is a||.||-closed subspace Z of X such that
X = Y ⊕ Z .
Theorem of Lindenstrauss, TzafririX Banach space.Every ||.||-closed subspace of X is complemented in X if, and only if, X isisomorphic to a Hilbert space.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 17 / 27
subspaces complemented
Definition
X Banach space.A ||.||-closed subspace Y of X is said to be complemented in X if there is a||.||-closed subspace Z of X such that
X = Y ⊕ Z .
Theorem of Lindenstrauss, TzafririX Banach space.Every ||.||-closed subspace of X is complemented in X if, and only if, X isisomorphic to a Hilbert space.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 17 / 27
subspaces complemented
DefinitionX Banach space.
A ||.||-closed subspace Y of X is said to be complemented in X if there is a||.||-closed subspace Z of X such that
X = Y ⊕ Z .
Theorem of Lindenstrauss, TzafririX Banach space.Every ||.||-closed subspace of X is complemented in X if, and only if, X isisomorphic to a Hilbert space.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 17 / 27
subspaces complemented
DefinitionX Banach space.A ||.||-closed subspace Y of X is said to be complemented in X
if there is a||.||-closed subspace Z of X such that
X = Y ⊕ Z .
Theorem of Lindenstrauss, TzafririX Banach space.Every ||.||-closed subspace of X is complemented in X if, and only if, X isisomorphic to a Hilbert space.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 17 / 27
subspaces complemented
DefinitionX Banach space.A ||.||-closed subspace Y of X is said to be complemented in X if there is a||.||-closed subspace Z of X
such that
X = Y ⊕ Z .
Theorem of Lindenstrauss, TzafririX Banach space.Every ||.||-closed subspace of X is complemented in X if, and only if, X isisomorphic to a Hilbert space.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 17 / 27
subspaces complemented
DefinitionX Banach space.A ||.||-closed subspace Y of X is said to be complemented in X if there is a||.||-closed subspace Z of X such that
X = Y ⊕ Z .
Theorem of Lindenstrauss, TzafririX Banach space.Every ||.||-closed subspace of X is complemented in X if, and only if, X isisomorphic to a Hilbert space.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 17 / 27
subspaces complemented
DefinitionX Banach space.A ||.||-closed subspace Y of X is said to be complemented in X if there is a||.||-closed subspace Z of X such that
X = Y ⊕ Z .
Theorem of Lindenstrauss, Tzafriri
X Banach space.Every ||.||-closed subspace of X is complemented in X if, and only if, X isisomorphic to a Hilbert space.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 17 / 27
subspaces complemented
DefinitionX Banach space.A ||.||-closed subspace Y of X is said to be complemented in X if there is a||.||-closed subspace Z of X such that
X = Y ⊕ Z .
Theorem of Lindenstrauss, TzafririX Banach space.
Every ||.||-closed subspace of X is complemented in X if, and only if, X isisomorphic to a Hilbert space.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 17 / 27
subspaces complemented
DefinitionX Banach space.A ||.||-closed subspace Y of X is said to be complemented in X if there is a||.||-closed subspace Z of X such that
X = Y ⊕ Z .
Theorem of Lindenstrauss, TzafririX Banach space.Every ||.||-closed subspace of X is complemented in X
if, and only if, X isisomorphic to a Hilbert space.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 17 / 27
subspaces complemented
DefinitionX Banach space.A ||.||-closed subspace Y of X is said to be complemented in X if there is a||.||-closed subspace Z of X such that
X = Y ⊕ Z .
Theorem of Lindenstrauss, TzafririX Banach space.Every ||.||-closed subspace of X is complemented in X if, and only if, X isisomorphic to a Hilbert space.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 17 / 27
subspaces complemented
DefinitionX Banach space.A ||.||-closed subspace Y of X is said to be complemented in X if there is a||.||-closed subspace Z of X such that
X = Y ⊕ Z .
Theorem of Lindenstrauss, TzafririX Banach space.Every ||.||-closed subspace of X is complemented in X if, and only if, X isisomorphic to a Hilbert space.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 17 / 27
Complemented algebra
A algebra, I ∈ IA.I is complemented if there exists J ∈ IA such that
A = I ⊕ J.
A is a complemented algebra if, every ideal I of A is complemented.
DefinitionA non null algebra A is simple lacks proper ideals.
TheoremA normed algebra with zero annihilator.A is complemented ⇐⇒ A is an algebra isomorphic to a direct sum of simplealgebras.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 18 / 27
Complemented algebra
A algebra, I ∈ IA.
I is complemented if there exists J ∈ IA such that
A = I ⊕ J.
A is a complemented algebra if, every ideal I of A is complemented.
DefinitionA non null algebra A is simple lacks proper ideals.
TheoremA normed algebra with zero annihilator.A is complemented ⇐⇒ A is an algebra isomorphic to a direct sum of simplealgebras.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 18 / 27
Complemented algebra
A algebra, I ∈ IA.I is complemented if there exists J ∈ IA such that
A = I ⊕ J.
A is a complemented algebra if, every ideal I of A is complemented.
DefinitionA non null algebra A is simple lacks proper ideals.
TheoremA normed algebra with zero annihilator.A is complemented ⇐⇒ A is an algebra isomorphic to a direct sum of simplealgebras.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 18 / 27
Complemented algebra
A algebra, I ∈ IA.I is complemented if there exists J ∈ IA such that
A = I ⊕ J.
A is a complemented algebra if, every ideal I of A is complemented.
DefinitionA non null algebra A is simple lacks proper ideals.
TheoremA normed algebra with zero annihilator.A is complemented ⇐⇒ A is an algebra isomorphic to a direct sum of simplealgebras.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 18 / 27
Complemented algebra
A algebra, I ∈ IA.I is complemented if there exists J ∈ IA such that
A = I ⊕ J.
A is a complemented algebra if, every ideal I of A is complemented.
DefinitionA non null algebra A is simple lacks proper ideals.
TheoremA normed algebra with zero annihilator.A is complemented ⇐⇒ A is an algebra isomorphic to a direct sum of simplealgebras.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 18 / 27
Complemented algebra
A algebra, I ∈ IA.I is complemented if there exists J ∈ IA such that
A = I ⊕ J.
A is a complemented algebra if, every ideal I of A is complemented.
Definition
A non null algebra A is simple lacks proper ideals.
TheoremA normed algebra with zero annihilator.A is complemented ⇐⇒ A is an algebra isomorphic to a direct sum of simplealgebras.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 18 / 27
Complemented algebra
A algebra, I ∈ IA.I is complemented if there exists J ∈ IA such that
A = I ⊕ J.
A is a complemented algebra if, every ideal I of A is complemented.
DefinitionA non null algebra A is simple lacks proper ideals.
TheoremA normed algebra with zero annihilator.A is complemented ⇐⇒ A is an algebra isomorphic to a direct sum of simplealgebras.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 18 / 27
Complemented algebra
A algebra, I ∈ IA.I is complemented if there exists J ∈ IA such that
A = I ⊕ J.
A is a complemented algebra if, every ideal I of A is complemented.
DefinitionA non null algebra A is simple lacks proper ideals.
Theorem
A normed algebra with zero annihilator.A is complemented ⇐⇒ A is an algebra isomorphic to a direct sum of simplealgebras.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 18 / 27
Complemented algebra
A algebra, I ∈ IA.I is complemented if there exists J ∈ IA such that
A = I ⊕ J.
A is a complemented algebra if, every ideal I of A is complemented.
DefinitionA non null algebra A is simple lacks proper ideals.
TheoremA normed algebra with zero annihilator.
A is complemented ⇐⇒ A is an algebra isomorphic to a direct sum of simplealgebras.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 18 / 27
Complemented algebra
A algebra, I ∈ IA.I is complemented if there exists J ∈ IA such that
A = I ⊕ J.
A is a complemented algebra if, every ideal I of A is complemented.
DefinitionA non null algebra A is simple lacks proper ideals.
TheoremA normed algebra with zero annihilator.A is complemented ⇐⇒
A is an algebra isomorphic to a direct sum of simplealgebras.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 18 / 27
Complemented algebra
A algebra, I ∈ IA.I is complemented if there exists J ∈ IA such that
A = I ⊕ J.
A is a complemented algebra if, every ideal I of A is complemented.
DefinitionA non null algebra A is simple lacks proper ideals.
TheoremA normed algebra with zero annihilator.A is complemented ⇐⇒ A is an algebra isomorphic to a direct sum of simplealgebras.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 18 / 27
∼-Complemented algebra
A algebra,
∼ closure operation on IA.A ∼-closed ideal I of A is ∼-complemented in A if there exists an ∼-closedideal J of A such that
A = I ⊕ J.
A is a ∼-complemented algebra when every ∼-closed ideal of A is∼-complemented.∼ is additive if I + J = I + J (I, J ∈ IA)
minimal caracter of the π-complementationEvery ∼-complemented algebra is π-complemented.
PropositionA algebra with Ann(A) = 0.A complemented =⇒ A ε-complemented =⇒ A π-complemented =⇒ A
semiprime
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 19 / 27
∼-Complemented algebra
A algebra, ∼ closure operation on IA.
A ∼-closed ideal I of A is ∼-complemented in A if there exists an ∼-closedideal J of A such that
A = I ⊕ J.
A is a ∼-complemented algebra when every ∼-closed ideal of A is∼-complemented.∼ is additive if I + J = I + J (I, J ∈ IA)
minimal caracter of the π-complementationEvery ∼-complemented algebra is π-complemented.
PropositionA algebra with Ann(A) = 0.A complemented =⇒ A ε-complemented =⇒ A π-complemented =⇒ A
semiprime
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 19 / 27
∼-Complemented algebra
A algebra, ∼ closure operation on IA.A ∼-closed ideal I of A is ∼-complemented in A if there exists an ∼-closedideal J of A
such thatA = I ⊕ J.
A is a ∼-complemented algebra when every ∼-closed ideal of A is∼-complemented.∼ is additive if I + J = I + J (I, J ∈ IA)
minimal caracter of the π-complementationEvery ∼-complemented algebra is π-complemented.
PropositionA algebra with Ann(A) = 0.A complemented =⇒ A ε-complemented =⇒ A π-complemented =⇒ A
semiprime
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 19 / 27
∼-Complemented algebra
A algebra, ∼ closure operation on IA.A ∼-closed ideal I of A is ∼-complemented in A if there exists an ∼-closedideal J of A such that
A = I ⊕ J.
A is a ∼-complemented algebra when every ∼-closed ideal of A is∼-complemented.∼ is additive if I + J = I + J (I, J ∈ IA)
minimal caracter of the π-complementationEvery ∼-complemented algebra is π-complemented.
PropositionA algebra with Ann(A) = 0.A complemented =⇒ A ε-complemented =⇒ A π-complemented =⇒ A
semiprime
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 19 / 27
∼-Complemented algebra
A algebra, ∼ closure operation on IA.A ∼-closed ideal I of A is ∼-complemented in A if there exists an ∼-closedideal J of A such that
A = I ⊕ J.
A is a ∼-complemented algebra when every ∼-closed ideal of A is∼-complemented.∼ is additive if I + J = I + J (I, J ∈ IA)
minimal caracter of the π-complementationEvery ∼-complemented algebra is π-complemented.
PropositionA algebra with Ann(A) = 0.A complemented =⇒ A ε-complemented =⇒ A π-complemented =⇒ A
semiprime
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 19 / 27
∼-Complemented algebra
A algebra, ∼ closure operation on IA.A ∼-closed ideal I of A is ∼-complemented in A if there exists an ∼-closedideal J of A such that
A = I ⊕ J.
A is a ∼-complemented algebra when every ∼-closed ideal of A is∼-complemented.
∼ is additive if I + J = I + J (I, J ∈ IA)
minimal caracter of the π-complementationEvery ∼-complemented algebra is π-complemented.
PropositionA algebra with Ann(A) = 0.A complemented =⇒ A ε-complemented =⇒ A π-complemented =⇒ A
semiprime
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 19 / 27
∼-Complemented algebra
A algebra, ∼ closure operation on IA.A ∼-closed ideal I of A is ∼-complemented in A if there exists an ∼-closedideal J of A such that
A = I ⊕ J.
A is a ∼-complemented algebra when every ∼-closed ideal of A is∼-complemented.∼ is additive if I + J = I + J (I, J ∈ IA)
minimal caracter of the π-complementationEvery ∼-complemented algebra is π-complemented.
PropositionA algebra with Ann(A) = 0.A complemented =⇒ A ε-complemented =⇒ A π-complemented =⇒ A
semiprime
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 19 / 27
∼-Complemented algebra
A algebra, ∼ closure operation on IA.A ∼-closed ideal I of A is ∼-complemented in A if there exists an ∼-closedideal J of A such that
A = I ⊕ J.
A is a ∼-complemented algebra when every ∼-closed ideal of A is∼-complemented.∼ is additive if I + J = I + J (I, J ∈ IA)
minimal caracter of the π-complementation
Every ∼-complemented algebra is π-complemented.
PropositionA algebra with Ann(A) = 0.A complemented =⇒ A ε-complemented =⇒ A π-complemented =⇒ A
semiprime
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 19 / 27
∼-Complemented algebra
A algebra, ∼ closure operation on IA.A ∼-closed ideal I of A is ∼-complemented in A if there exists an ∼-closedideal J of A such that
A = I ⊕ J.
A is a ∼-complemented algebra when every ∼-closed ideal of A is∼-complemented.∼ is additive if I + J = I + J (I, J ∈ IA)
minimal caracter of the π-complementationEvery ∼-complemented algebra is π-complemented.
PropositionA algebra with Ann(A) = 0.A complemented =⇒ A ε-complemented =⇒ A π-complemented =⇒ A
semiprime
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 19 / 27
∼-Complemented algebra
A algebra, ∼ closure operation on IA.A ∼-closed ideal I of A is ∼-complemented in A if there exists an ∼-closedideal J of A such that
A = I ⊕ J.
A is a ∼-complemented algebra when every ∼-closed ideal of A is∼-complemented.∼ is additive if I + J = I + J (I, J ∈ IA)
minimal caracter of the π-complementationEvery ∼-complemented algebra is π-complemented.
PropositionA algebra with Ann(A) = 0.A complemented =⇒ A ε-complemented =⇒ A π-complemented =⇒ A
semiprime
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 19 / 27
ε-complemented algebra not complementedThe algebra ∞ of all bounded complex sequences.
π-complemented algebra not ε-complementedAlgebra of Albert
PropositionA algebra with Ann(A) = 0.A ε-complementend ⇐⇒ A m.s.p. and π-complemented.
theoremA algebraA is π-complemented ⇐⇒ A is semiprime and the π-closure is additive
corollaryA algebra with zero annihilator.A ε-complemented ⇐⇒ A m.s.p. and the ε-closure is additive
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 20 / 27
ε-complemented algebra not complemented
The algebra ∞ of all bounded complex sequences.
π-complemented algebra not ε-complementedAlgebra of Albert
PropositionA algebra with Ann(A) = 0.A ε-complementend ⇐⇒ A m.s.p. and π-complemented.
theoremA algebraA is π-complemented ⇐⇒ A is semiprime and the π-closure is additive
corollaryA algebra with zero annihilator.A ε-complemented ⇐⇒ A m.s.p. and the ε-closure is additive
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 20 / 27
ε-complemented algebra not complementedThe algebra ∞ of all bounded complex sequences.
π-complemented algebra not ε-complementedAlgebra of Albert
PropositionA algebra with Ann(A) = 0.A ε-complementend ⇐⇒ A m.s.p. and π-complemented.
theoremA algebraA is π-complemented ⇐⇒ A is semiprime and the π-closure is additive
corollaryA algebra with zero annihilator.A ε-complemented ⇐⇒ A m.s.p. and the ε-closure is additive
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 20 / 27
ε-complemented algebra not complementedThe algebra ∞ of all bounded complex sequences.
π-complemented algebra not ε-complemented
Algebra of Albert
PropositionA algebra with Ann(A) = 0.A ε-complementend ⇐⇒ A m.s.p. and π-complemented.
theoremA algebraA is π-complemented ⇐⇒ A is semiprime and the π-closure is additive
corollaryA algebra with zero annihilator.A ε-complemented ⇐⇒ A m.s.p. and the ε-closure is additive
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 20 / 27
ε-complemented algebra not complementedThe algebra ∞ of all bounded complex sequences.
π-complemented algebra not ε-complementedAlgebra of Albert
PropositionA algebra with Ann(A) = 0.A ε-complementend ⇐⇒ A m.s.p. and π-complemented.
theoremA algebraA is π-complemented ⇐⇒ A is semiprime and the π-closure is additive
corollaryA algebra with zero annihilator.A ε-complemented ⇐⇒ A m.s.p. and the ε-closure is additive
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 20 / 27
ε-complemented algebra not complementedThe algebra ∞ of all bounded complex sequences.
π-complemented algebra not ε-complementedAlgebra of Albert
PropositionA algebra with Ann(A) = 0.
A ε-complementend ⇐⇒ A m.s.p. and π-complemented.
theoremA algebraA is π-complemented ⇐⇒ A is semiprime and the π-closure is additive
corollaryA algebra with zero annihilator.A ε-complemented ⇐⇒ A m.s.p. and the ε-closure is additive
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 20 / 27
ε-complemented algebra not complementedThe algebra ∞ of all bounded complex sequences.
π-complemented algebra not ε-complementedAlgebra of Albert
PropositionA algebra with Ann(A) = 0.A ε-complementend ⇐⇒ A m.s.p. and π-complemented.
theoremA algebraA is π-complemented ⇐⇒ A is semiprime and the π-closure is additive
corollaryA algebra with zero annihilator.A ε-complemented ⇐⇒ A m.s.p. and the ε-closure is additive
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 20 / 27
ε-complemented algebra not complementedThe algebra ∞ of all bounded complex sequences.
π-complemented algebra not ε-complementedAlgebra of Albert
PropositionA algebra with Ann(A) = 0.A ε-complementend ⇐⇒ A m.s.p. and π-complemented.
theoremA algebraA is π-complemented ⇐⇒ A is semiprime and the π-closure is additive
corollaryA algebra with zero annihilator.A ε-complemented ⇐⇒ A m.s.p. and the ε-closure is additive
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 20 / 27
ε-complemented algebra not complementedThe algebra ∞ of all bounded complex sequences.
π-complemented algebra not ε-complementedAlgebra of Albert
PropositionA algebra with Ann(A) = 0.A ε-complementend ⇐⇒ A m.s.p. and π-complemented.
theoremA algebraA is π-complemented ⇐⇒ A is semiprime and the π-closure is additive
corollaryA algebra with zero annihilator.
A ε-complemented ⇐⇒ A m.s.p. and the ε-closure is additive
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 20 / 27
ε-complemented algebra not complementedThe algebra ∞ of all bounded complex sequences.
π-complemented algebra not ε-complementedAlgebra of Albert
PropositionA algebra with Ann(A) = 0.A ε-complementend ⇐⇒ A m.s.p. and π-complemented.
theoremA algebraA is π-complemented ⇐⇒ A is semiprime and the π-closure is additive
corollaryA algebra with zero annihilator.A ε-complemented ⇐⇒ A m.s.p. and the ε-closure is additive
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 20 / 27
ε-complemented algebra not complementedThe algebra ∞ of all bounded complex sequences.
π-complemented algebra not ε-complementedAlgebra of Albert
PropositionA algebra with Ann(A) = 0.A ε-complementend ⇐⇒ A m.s.p. and π-complemented.
theoremA algebraA is π-complemented ⇐⇒ A is semiprime and the π-closure is additive
corollaryA algebra with zero annihilator.A ε-complemented ⇐⇒ A m.s.p. and the ε-closure is additive
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 20 / 27
Decomposable algebras
DefinitionA algebra, ∼ closure operation on IA, m∼
A= minimal elements of I∼
A.
A is a ∼-decomposable algebra whenever A = B∈m∼
A
B
M∼A
= maximal elements ofI∼A,
A is a ∼-radical algebra whenever M∼A
= ∅.
TheoremA algebra. T.F.A.E.
A is m.s.p.A = A0 ⊕ A1, where A0 is a π-radical m.s.p. algebra and a A1 is aπ-decomposable m.s.p. algebra.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 21 / 27
Decomposable algebras
DefinitionA algebra, ∼ closure operation on IA, m∼
A= minimal elements of I∼
A.
A is a ∼-decomposable algebra whenever A = B∈m∼
A
B
M∼A
= maximal elements ofI∼A,
A is a ∼-radical algebra whenever M∼A
= ∅.
TheoremA algebra. T.F.A.E.
A is m.s.p.A = A0 ⊕ A1, where A0 is a π-radical m.s.p. algebra and a A1 is aπ-decomposable m.s.p. algebra.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 21 / 27
Decomposable algebras
DefinitionA algebra, ∼ closure operation on IA, m∼
A= minimal elements of I∼
A.
A is a ∼-decomposable algebra whenever A = B∈m∼
A
B
M∼A
= maximal elements ofI∼A,
A is a ∼-radical algebra whenever M∼A
= ∅.
TheoremA algebra. T.F.A.E.
A is m.s.p.A = A0 ⊕ A1, where A0 is a π-radical m.s.p. algebra and a A1 is aπ-decomposable m.s.p. algebra.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 21 / 27
Decomposable algebras
DefinitionA algebra, ∼ closure operation on IA, m∼
A= minimal elements of I∼
A.
A is a ∼-decomposable algebra whenever A = B∈m∼
A
B
M∼A
= maximal elements ofI∼A,
A is a ∼-radical algebra whenever M∼A
= ∅.
TheoremA algebra. T.F.A.E.
A is m.s.p.A = A0 ⊕ A1, where A0 is a π-radical m.s.p. algebra and a A1 is aπ-decomposable m.s.p. algebra.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 21 / 27
Decomposable algebras
DefinitionA algebra, ∼ closure operation on IA, m∼
A= minimal elements of I∼
A.
A is a ∼-decomposable algebra whenever A = B∈m∼
A
B
M∼A
= maximal elements ofI∼A,
A is a ∼-radical algebra whenever M∼A
= ∅.
TheoremA algebra. T.F.A.E.
A is m.s.p.A = A0 ⊕ A1, where A0 is a π-radical m.s.p. algebra and a A1 is aπ-decomposable m.s.p. algebra.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 21 / 27
Decomposable algebras
DefinitionA algebra, ∼ closure operation on IA, m∼
A= minimal elements of I∼
A.
A is a ∼-decomposable algebra whenever A = B∈m∼
A
B
M∼A
= maximal elements ofI∼A,
A is a ∼-radical algebra whenever M∼A
= ∅.
TheoremA algebra. T.F.A.E.
A is m.s.p.A = A0 ⊕ A1, where A0 is a π-radical m.s.p. algebra and a A1 is aπ-decomposable m.s.p. algebra.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 21 / 27
Decomposable algebras
DefinitionA algebra, ∼ closure operation on IA, m∼
A= minimal elements of I∼
A.
A is a ∼-decomposable algebra whenever A = B∈m∼
A
B
M∼A
= maximal elements ofI∼A,
A is a ∼-radical algebra whenever M∼A
= ∅.
TheoremA algebra. T.F.A.E.
A is m.s.p.A = A0 ⊕ A1, where A0 is a π-radical m.s.p. algebra and a A1 is aπ-decomposable m.s.p. algebra.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 21 / 27
Decomposable algebras
DefinitionA algebra, ∼ closure operation on IA, m∼
A= minimal elements of I∼
A.
A is a ∼-decomposable algebra whenever A = B∈m∼
A
B
M∼A
= maximal elements ofI∼A,
A is a ∼-radical algebra whenever M∼A
= ∅.
TheoremA algebra. T.F.A.E.
A is m.s.p.
A = A0 ⊕ A1, where A0 is a π-radical m.s.p. algebra and a A1 is aπ-decomposable m.s.p. algebra.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 21 / 27
Decomposable algebras
DefinitionA algebra, ∼ closure operation on IA, m∼
A= minimal elements of I∼
A.
A is a ∼-decomposable algebra whenever A = B∈m∼
A
B
M∼A
= maximal elements ofI∼A,
A is a ∼-radical algebra whenever M∼A
= ∅.
TheoremA algebra. T.F.A.E.
A is m.s.p.A = A0 ⊕ A1, where A0 is a π-radical m.s.p. algebra and a A1 is aπ-decomposable m.s.p. algebra.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 21 / 27
DefinitionA is ∼-atomic if each nonzero ∼-closed ideal of A contains a minimal∼-closed ideal.
DefinitionA is m.p. ⇐⇒ both A and M(A) are prime algebras
Theorem.A algebra with zero annihilator,
A ε-decomposable ⇐⇒ A ε-atomic m.s.p. algebra.Moreover, in the case, every minimal ε-closed ideal of A is an m.p. algebra.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 22 / 27
DefinitionA is ∼-atomic if each nonzero ∼-closed ideal of A contains a minimal∼-closed ideal.
DefinitionA is m.p. ⇐⇒ both A and M(A) are prime algebras
Theorem.A algebra with zero annihilator,
A ε-decomposable ⇐⇒ A ε-atomic m.s.p. algebra.Moreover, in the case, every minimal ε-closed ideal of A is an m.p. algebra.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 22 / 27
DefinitionA is ∼-atomic if each nonzero ∼-closed ideal of A contains a minimal∼-closed ideal.
DefinitionA is m.p. ⇐⇒ both A and M(A) are prime algebras
Theorem.
A algebra with zero annihilator,A ε-decomposable ⇐⇒ A ε-atomic m.s.p. algebra.
Moreover, in the case, every minimal ε-closed ideal of A is an m.p. algebra.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 22 / 27
DefinitionA is ∼-atomic if each nonzero ∼-closed ideal of A contains a minimal∼-closed ideal.
DefinitionA is m.p. ⇐⇒ both A and M(A) are prime algebras
Theorem.A algebra with zero annihilator,
A ε-decomposable ⇐⇒ A ε-atomic m.s.p. algebra.Moreover, in the case, every minimal ε-closed ideal of A is an m.p. algebra.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 22 / 27
DefinitionA is ∼-atomic if each nonzero ∼-closed ideal of A contains a minimal∼-closed ideal.
DefinitionA is m.p. ⇐⇒ both A and M(A) are prime algebras
Theorem.A algebra with zero annihilator,
A ε-decomposable ⇐⇒ A ε-atomic m.s.p. algebra.
Moreover, in the case, every minimal ε-closed ideal of A is an m.p. algebra.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 22 / 27
DefinitionA is ∼-atomic if each nonzero ∼-closed ideal of A contains a minimal∼-closed ideal.
DefinitionA is m.p. ⇐⇒ both A and M(A) are prime algebras
Theorem.A algebra with zero annihilator,
A ε-decomposable ⇐⇒ A ε-atomic m.s.p. algebra.Moreover, in the case, every minimal ε-closed ideal of A is an m.p. algebra.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 22 / 27
DefinitionA is ∼-atomic if each nonzero ∼-closed ideal of A contains a minimal∼-closed ideal.
DefinitionA is m.p. ⇐⇒ both A and M(A) are prime algebras
Theorem.A algebra with zero annihilator,
A ε-decomposable ⇐⇒ A ε-atomic m.s.p. algebra.Moreover, in the case, every minimal ε-closed ideal of A is an m.p. algebra.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 22 / 27
A normed m.s.p. algebra
||.||-atomic =⇒ ε-atomic
The converse is not true
ExampleThe algebra P of all polynomials is an (ε-atomic) m.p. commutativeassociative algebra, but if P is endowed with the norm
||p|| = Max|p(t)|; t ∈ [0, 1],
then lacks minimal ||.||-closed ideals.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 23 / 27
A normed m.s.p. algebra||.||-atomic =⇒ ε-atomic
The converse is not true
ExampleThe algebra P of all polynomials is an (ε-atomic) m.p. commutativeassociative algebra, but if P is endowed with the norm
||p|| = Max|p(t)|; t ∈ [0, 1],
then lacks minimal ||.||-closed ideals.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 23 / 27
A normed m.s.p. algebra||.||-atomic =⇒ ε-atomic
The converse is not true
ExampleThe algebra P of all polynomials is an (ε-atomic) m.p. commutativeassociative algebra, but if P is endowed with the norm
||p|| = Max|p(t)|; t ∈ [0, 1],
then lacks minimal ||.||-closed ideals.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 23 / 27
A normed m.s.p. algebra||.||-atomic =⇒ ε-atomic
The converse is not true
ExampleThe algebra P of all polynomials is an (ε-atomic) m.p. commutativeassociative algebra, but
if P is endowed with the norm
||p|| = Max|p(t)|; t ∈ [0, 1],
then lacks minimal ||.||-closed ideals.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 23 / 27
A normed m.s.p. algebra||.||-atomic =⇒ ε-atomic
The converse is not true
ExampleThe algebra P of all polynomials is an (ε-atomic) m.p. commutativeassociative algebra, but if P is endowed with the norm
||p|| = Max|p(t)|; t ∈ [0, 1],
then lacks minimal ||.||-closed ideals.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 23 / 27
A normed m.s.p. algebra||.||-atomic =⇒ ε-atomic
The converse is not true
ExampleThe algebra P of all polynomials is an (ε-atomic) m.p. commutativeassociative algebra, but if P is endowed with the norm
||p|| = Max|p(t)|; t ∈ [0, 1],
then lacks minimal ||.||-closed ideals.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 23 / 27
TheoremA normed m.s.p. algebra.
A ||.||-atomic.
A ε-atomic and mεA
= I; I ∈ m||.||A
TheoremA normed algebra with zero annihilator. T.F.A.E.
A ||.||-decomposable
A ||.||-atomic and A = I ⊕ Ann(I)||.||
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 24 / 27
TheoremA normed m.s.p. algebra.
A ||.||-atomic.
A ε-atomic and mεA
= I; I ∈ m||.||A
TheoremA normed algebra with zero annihilator. T.F.A.E.
A ||.||-decomposable
A ||.||-atomic and A = I ⊕ Ann(I)||.||
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 24 / 27
TheoremA normed m.s.p. algebra.
A ||.||-atomic.
A ε-atomic and mεA
= I; I ∈ m||.||A
TheoremA normed algebra with zero annihilator. T.F.A.E.
A ||.||-decomposable
A ||.||-atomic and A = I ⊕ Ann(I)||.||
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 24 / 27
TheoremA normed m.s.p. algebra.
A ||.||-atomic.
A ε-atomic and mεA
= I; I ∈ m||.||A
TheoremA normed algebra with zero annihilator. T.F.A.E.
A ||.||-decomposable
A ||.||-atomic and A = I ⊕ Ann(I)||.||
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 24 / 27
TheoremA normed m.s.p. algebra.
A ||.||-atomic.
A ε-atomic and mεA
= I; I ∈ m||.||A
Theorem
A normed algebra with zero annihilator. T.F.A.E.
A ||.||-decomposable
A ||.||-atomic and A = I ⊕ Ann(I)||.||
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 24 / 27
TheoremA normed m.s.p. algebra.
A ||.||-atomic.
A ε-atomic and mεA
= I; I ∈ m||.||A
TheoremA normed algebra with zero annihilator. T.F.A.E.
A ||.||-decomposable
A ||.||-atomic and A = I ⊕ Ann(I)||.||
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 24 / 27
TheoremA normed m.s.p. algebra.
A ||.||-atomic.
A ε-atomic and mεA
= I; I ∈ m||.||A
TheoremA normed algebra with zero annihilator. T.F.A.E.
A ||.||-decomposable
A ||.||-atomic and A = I ⊕ Ann(I)||.||
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 24 / 27
TheoremA normed m.s.p. algebra.
A ||.||-atomic.
A ε-atomic and mεA
= I; I ∈ m||.||A
TheoremA normed algebra with zero annihilator. T.F.A.E.
A ||.||-decomposable
A ||.||-atomic and A = I ⊕ Ann(I)||.||
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 24 / 27
TheoremA normed m.s.p. algebra.
A ||.||-atomic.
A ε-atomic and mεA
= I; I ∈ m||.||A
TheoremA normed algebra with zero annihilator. T.F.A.E.
A ||.||-decomposable
A ||.||-atomic and A = I ⊕ Ann(I)||.||
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 24 / 27
Proposition
Ω locally compact topological space.C0(Ω) is ||.||-atomic ⇐⇒ C0(Ω) is ε-atomic ⇐⇒ The set of all isolated points ofΩ is dense in Ω.
Corollary
The algebra C([0, 1]) of all complex-valued continuous functions on theinterval [0, 1] is a non-ε-atomic m.s.p. algebra.The algebra c of all convergent complex sequences, and the algebra ∞of all bounded complex sequences are ε-decomposable algebras.The algebra c0 of all null complex sequences is ||.||-decomposable.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 25 / 27
PropositionΩ locally compact topological space.
C0(Ω) is ||.||-atomic ⇐⇒ C0(Ω) is ε-atomic ⇐⇒ The set of all isolated points ofΩ is dense in Ω.
Corollary
The algebra C([0, 1]) of all complex-valued continuous functions on theinterval [0, 1] is a non-ε-atomic m.s.p. algebra.The algebra c of all convergent complex sequences, and the algebra ∞of all bounded complex sequences are ε-decomposable algebras.The algebra c0 of all null complex sequences is ||.||-decomposable.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 25 / 27
PropositionΩ locally compact topological space.C0(Ω) is ||.||-atomic
⇐⇒ C0(Ω) is ε-atomic ⇐⇒ The set of all isolated points ofΩ is dense in Ω.
Corollary
The algebra C([0, 1]) of all complex-valued continuous functions on theinterval [0, 1] is a non-ε-atomic m.s.p. algebra.The algebra c of all convergent complex sequences, and the algebra ∞of all bounded complex sequences are ε-decomposable algebras.The algebra c0 of all null complex sequences is ||.||-decomposable.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 25 / 27
PropositionΩ locally compact topological space.C0(Ω) is ||.||-atomic ⇐⇒ C0(Ω) is ε-atomic
⇐⇒ The set of all isolated points ofΩ is dense in Ω.
Corollary
The algebra C([0, 1]) of all complex-valued continuous functions on theinterval [0, 1] is a non-ε-atomic m.s.p. algebra.The algebra c of all convergent complex sequences, and the algebra ∞of all bounded complex sequences are ε-decomposable algebras.The algebra c0 of all null complex sequences is ||.||-decomposable.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 25 / 27
PropositionΩ locally compact topological space.C0(Ω) is ||.||-atomic ⇐⇒ C0(Ω) is ε-atomic ⇐⇒ The set of all isolated points ofΩ is dense in Ω.
Corollary
The algebra C([0, 1]) of all complex-valued continuous functions on theinterval [0, 1] is a non-ε-atomic m.s.p. algebra.The algebra c of all convergent complex sequences, and the algebra ∞of all bounded complex sequences are ε-decomposable algebras.The algebra c0 of all null complex sequences is ||.||-decomposable.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 25 / 27
PropositionΩ locally compact topological space.C0(Ω) is ||.||-atomic ⇐⇒ C0(Ω) is ε-atomic ⇐⇒ The set of all isolated points ofΩ is dense in Ω.
Corollary
The algebra C([0, 1]) of all complex-valued continuous functions on theinterval [0, 1] is a non-ε-atomic m.s.p. algebra.The algebra c of all convergent complex sequences, and the algebra ∞of all bounded complex sequences are ε-decomposable algebras.The algebra c0 of all null complex sequences is ||.||-decomposable.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 25 / 27
PropositionΩ locally compact topological space.C0(Ω) is ||.||-atomic ⇐⇒ C0(Ω) is ε-atomic ⇐⇒ The set of all isolated points ofΩ is dense in Ω.
CorollaryThe algebra C([0, 1]) of all complex-valued continuous functions on theinterval [0, 1]
is a non-ε-atomic m.s.p. algebra.
The algebra c of all convergent complex sequences, and the algebra ∞of all bounded complex sequences are ε-decomposable algebras.The algebra c0 of all null complex sequences is ||.||-decomposable.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 25 / 27
PropositionΩ locally compact topological space.C0(Ω) is ||.||-atomic ⇐⇒ C0(Ω) is ε-atomic ⇐⇒ The set of all isolated points ofΩ is dense in Ω.
CorollaryThe algebra C([0, 1]) of all complex-valued continuous functions on theinterval [0, 1]
is a non-ε-atomic m.s.p. algebra.
The algebra c of all convergent complex sequences, and the algebra ∞of all bounded complex sequences are ε-decomposable algebras.The algebra c0 of all null complex sequences is ||.||-decomposable.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 25 / 27
PropositionΩ locally compact topological space.C0(Ω) is ||.||-atomic ⇐⇒ C0(Ω) is ε-atomic ⇐⇒ The set of all isolated points ofΩ is dense in Ω.
CorollaryThe algebra C([0, 1]) of all complex-valued continuous functions on theinterval [0, 1] is a non-ε-atomic m.s.p. algebra.The algebra c of all convergent complex sequences,
and the algebra ∞of all bounded complex sequences are ε-decomposable algebras.
The algebra c0 of all null complex sequences is ||.||-decomposable.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 25 / 27
PropositionΩ locally compact topological space.C0(Ω) is ||.||-atomic ⇐⇒ C0(Ω) is ε-atomic ⇐⇒ The set of all isolated points ofΩ is dense in Ω.
CorollaryThe algebra C([0, 1]) of all complex-valued continuous functions on theinterval [0, 1] is a non-ε-atomic m.s.p. algebra.The algebra c of all convergent complex sequences,
and the algebra ∞of all bounded complex sequences are ε-decomposable algebras.
The algebra c0 of all null complex sequences is ||.||-decomposable.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 25 / 27
PropositionΩ locally compact topological space.C0(Ω) is ||.||-atomic ⇐⇒ C0(Ω) is ε-atomic ⇐⇒ The set of all isolated points ofΩ is dense in Ω.
CorollaryThe algebra C([0, 1]) of all complex-valued continuous functions on theinterval [0, 1] is a non-ε-atomic m.s.p. algebra.The algebra c of all convergent complex sequences, and the algebra ∞of all bounded complex sequences
are ε-decomposable algebras.
The algebra c0 of all null complex sequences is ||.||-decomposable.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 25 / 27
PropositionΩ locally compact topological space.C0(Ω) is ||.||-atomic ⇐⇒ C0(Ω) is ε-atomic ⇐⇒ The set of all isolated points ofΩ is dense in Ω.
CorollaryThe algebra C([0, 1]) of all complex-valued continuous functions on theinterval [0, 1] is a non-ε-atomic m.s.p. algebra.The algebra c of all convergent complex sequences, and the algebra ∞of all bounded complex sequences are ε-decomposable algebras.
The algebra c0 of all null complex sequences is ||.||-decomposable.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 25 / 27
PropositionΩ locally compact topological space.C0(Ω) is ||.||-atomic ⇐⇒ C0(Ω) is ε-atomic ⇐⇒ The set of all isolated points ofΩ is dense in Ω.
CorollaryThe algebra C([0, 1]) of all complex-valued continuous functions on theinterval [0, 1] is a non-ε-atomic m.s.p. algebra.The algebra c of all convergent complex sequences, and the algebra ∞of all bounded complex sequences are ε-decomposable algebras.The algebra c0 of all null complex sequences is ||.||-decomposable.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 25 / 27
Finite dimension
Jacobson´s TheoremA algebra. T.F.A.E.
A is finite dimensional and M(A) is semiprime;A = ⊕n
i=0Bi is a direct sum of ideals, one of them, say B0, is a finitedimensional null algebra and the others are finite dimensional simplealgebras.
CorollaryA algebraA finite-dimensional m.s.p. algebra ⇐⇒ A is isomorphic to a direct sum offinitely many finite-dimensional simple algebras.
TheoremA finite dimensional algebra. A complemented ⇐⇒ A ε-complemented ⇐⇒M(A) semiprime ⇐⇒ M(A) π-complemented ⇐⇒ M(A) complemented.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 26 / 27
Finite dimension
Jacobson´s Theorem
A algebra. T.F.A.E.
A is finite dimensional and M(A) is semiprime;A = ⊕n
i=0Bi is a direct sum of ideals, one of them, say B0, is a finitedimensional null algebra and the others are finite dimensional simplealgebras.
CorollaryA algebraA finite-dimensional m.s.p. algebra ⇐⇒ A is isomorphic to a direct sum offinitely many finite-dimensional simple algebras.
TheoremA finite dimensional algebra. A complemented ⇐⇒ A ε-complemented ⇐⇒M(A) semiprime ⇐⇒ M(A) π-complemented ⇐⇒ M(A) complemented.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 26 / 27
Finite dimension
Jacobson´s TheoremA algebra. T.F.A.E.
A is finite dimensional and M(A) is semiprime;A = ⊕n
i=0Bi is a direct sum of ideals, one of them, say B0, is a finitedimensional null algebra and the others are finite dimensional simplealgebras.
CorollaryA algebraA finite-dimensional m.s.p. algebra ⇐⇒ A is isomorphic to a direct sum offinitely many finite-dimensional simple algebras.
TheoremA finite dimensional algebra. A complemented ⇐⇒ A ε-complemented ⇐⇒M(A) semiprime ⇐⇒ M(A) π-complemented ⇐⇒ M(A) complemented.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 26 / 27
Finite dimension
Jacobson´s TheoremA algebra. T.F.A.E.
A is finite dimensional and M(A) is semiprime;
A = ⊕n
i=0Bi is a direct sum of ideals,
one of them, say B0, is a finitedimensional null algebra and the others are finite dimensional simplealgebras.
CorollaryA algebraA finite-dimensional m.s.p. algebra ⇐⇒ A is isomorphic to a direct sum offinitely many finite-dimensional simple algebras.
TheoremA finite dimensional algebra. A complemented ⇐⇒ A ε-complemented ⇐⇒M(A) semiprime ⇐⇒ M(A) π-complemented ⇐⇒ M(A) complemented.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 26 / 27
Finite dimension
Jacobson´s TheoremA algebra. T.F.A.E.
A is finite dimensional and M(A) is semiprime;A = ⊕n
i=0Bi is a direct sum of ideals,
one of them, say B0, is a finitedimensional null algebra and the others are finite dimensional simplealgebras.
CorollaryA algebraA finite-dimensional m.s.p. algebra ⇐⇒ A is isomorphic to a direct sum offinitely many finite-dimensional simple algebras.
TheoremA finite dimensional algebra. A complemented ⇐⇒ A ε-complemented ⇐⇒M(A) semiprime ⇐⇒ M(A) π-complemented ⇐⇒ M(A) complemented.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 26 / 27
Finite dimension
Jacobson´s TheoremA algebra. T.F.A.E.
A is finite dimensional and M(A) is semiprime;A = ⊕n
i=0Bi is a direct sum of ideals,
one of them, say B0, is a finitedimensional null algebra and the others are finite dimensional simplealgebras.
CorollaryA algebraA finite-dimensional m.s.p. algebra ⇐⇒ A is isomorphic to a direct sum offinitely many finite-dimensional simple algebras.
TheoremA finite dimensional algebra. A complemented ⇐⇒ A ε-complemented ⇐⇒M(A) semiprime ⇐⇒ M(A) π-complemented ⇐⇒ M(A) complemented.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 26 / 27
Finite dimension
Jacobson´s TheoremA algebra. T.F.A.E.
A is finite dimensional and M(A) is semiprime;A = ⊕n
i=0Bi is a direct sum of ideals, one of them, say B0, is a finitedimensional null algebra
and the others are finite dimensional simplealgebras.
CorollaryA algebraA finite-dimensional m.s.p. algebra ⇐⇒ A is isomorphic to a direct sum offinitely many finite-dimensional simple algebras.
TheoremA finite dimensional algebra. A complemented ⇐⇒ A ε-complemented ⇐⇒M(A) semiprime ⇐⇒ M(A) π-complemented ⇐⇒ M(A) complemented.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 26 / 27
Finite dimension
Jacobson´s TheoremA algebra. T.F.A.E.
A is finite dimensional and M(A) is semiprime;A = ⊕n
i=0Bi is a direct sum of ideals, one of them, say B0, is a finitedimensional null algebra and the others are finite dimensional simplealgebras.
CorollaryA algebraA finite-dimensional m.s.p. algebra ⇐⇒ A is isomorphic to a direct sum offinitely many finite-dimensional simple algebras.
TheoremA finite dimensional algebra. A complemented ⇐⇒ A ε-complemented ⇐⇒M(A) semiprime ⇐⇒ M(A) π-complemented ⇐⇒ M(A) complemented.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 26 / 27
Finite dimension
Jacobson´s TheoremA algebra. T.F.A.E.
A is finite dimensional and M(A) is semiprime;A = ⊕n
i=0Bi is a direct sum of ideals, one of them, say B0, is a finitedimensional null algebra and the others are finite dimensional simplealgebras.
CorollaryA algebraA finite-dimensional m.s.p. algebra ⇐⇒ A is isomorphic to a direct sum offinitely many finite-dimensional simple algebras.
TheoremA finite dimensional algebra. A complemented ⇐⇒ A ε-complemented ⇐⇒M(A) semiprime ⇐⇒ M(A) π-complemented ⇐⇒ M(A) complemented.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 26 / 27
Finite dimension
Jacobson´s TheoremA algebra. T.F.A.E.
A is finite dimensional and M(A) is semiprime;A = ⊕n
i=0Bi is a direct sum of ideals, one of them, say B0, is a finitedimensional null algebra and the others are finite dimensional simplealgebras.
CorollaryA algebraA finite-dimensional m.s.p. algebra ⇐⇒ A is isomorphic to a direct sum offinitely many finite-dimensional simple algebras.
TheoremA finite dimensional algebra. A complemented ⇐⇒ A ε-complemented ⇐⇒M(A) semiprime ⇐⇒ M(A) π-complemented ⇐⇒ M(A) complemented.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 26 / 27
Finite dimension
Jacobson´s TheoremA algebra. T.F.A.E.
A is finite dimensional and M(A) is semiprime;A = ⊕n
i=0Bi is a direct sum of ideals, one of them, say B0, is a finitedimensional null algebra and the others are finite dimensional simplealgebras.
CorollaryA algebraA finite-dimensional m.s.p. algebra ⇐⇒ A is isomorphic to a direct sum offinitely many finite-dimensional simple algebras.
Theorem
A finite dimensional algebra. A complemented ⇐⇒ A ε-complemented ⇐⇒M(A) semiprime ⇐⇒ M(A) π-complemented ⇐⇒ M(A) complemented.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 26 / 27
Finite dimension
Jacobson´s TheoremA algebra. T.F.A.E.
A is finite dimensional and M(A) is semiprime;A = ⊕n
i=0Bi is a direct sum of ideals, one of them, say B0, is a finitedimensional null algebra and the others are finite dimensional simplealgebras.
CorollaryA algebraA finite-dimensional m.s.p. algebra ⇐⇒ A is isomorphic to a direct sum offinitely many finite-dimensional simple algebras.
TheoremA finite dimensional algebra.
A complemented ⇐⇒ A ε-complemented ⇐⇒M(A) semiprime ⇐⇒ M(A) π-complemented ⇐⇒ M(A) complemented.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 26 / 27
Finite dimension
Jacobson´s TheoremA algebra. T.F.A.E.
A is finite dimensional and M(A) is semiprime;A = ⊕n
i=0Bi is a direct sum of ideals, one of them, say B0, is a finitedimensional null algebra and the others are finite dimensional simplealgebras.
CorollaryA algebraA finite-dimensional m.s.p. algebra ⇐⇒ A is isomorphic to a direct sum offinitely many finite-dimensional simple algebras.
TheoremA finite dimensional algebra. A complemented ⇐⇒ A ε-complemented ⇐⇒
M(A) semiprime ⇐⇒ M(A) π-complemented ⇐⇒ M(A) complemented.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 26 / 27
Finite dimension
Jacobson´s TheoremA algebra. T.F.A.E.
A is finite dimensional and M(A) is semiprime;A = ⊕n
i=0Bi is a direct sum of ideals, one of them, say B0, is a finitedimensional null algebra and the others are finite dimensional simplealgebras.
CorollaryA algebraA finite-dimensional m.s.p. algebra ⇐⇒ A is isomorphic to a direct sum offinitely many finite-dimensional simple algebras.
TheoremA finite dimensional algebra. A complemented ⇐⇒ A ε-complemented ⇐⇒M(A) semiprime ⇐⇒
M(A) π-complemented ⇐⇒ M(A) complemented.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 26 / 27
Finite dimension
Jacobson´s TheoremA algebra. T.F.A.E.
A is finite dimensional and M(A) is semiprime;A = ⊕n
i=0Bi is a direct sum of ideals, one of them, say B0, is a finitedimensional null algebra and the others are finite dimensional simplealgebras.
CorollaryA algebraA finite-dimensional m.s.p. algebra ⇐⇒ A is isomorphic to a direct sum offinitely many finite-dimensional simple algebras.
TheoremA finite dimensional algebra. A complemented ⇐⇒ A ε-complemented ⇐⇒M(A) semiprime ⇐⇒ M(A) π-complemented ⇐⇒ M(A) complemented.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 26 / 27
Finite dimension
Jacobson´s TheoremA algebra. T.F.A.E.
A is finite dimensional and M(A) is semiprime;A = ⊕n
i=0Bi is a direct sum of ideals, one of them, say B0, is a finitedimensional null algebra and the others are finite dimensional simplealgebras.
CorollaryA algebraA finite-dimensional m.s.p. algebra ⇐⇒ A is isomorphic to a direct sum offinitely many finite-dimensional simple algebras.
TheoremA finite dimensional algebra. A complemented ⇐⇒ A ε-complemented ⇐⇒M(A) semiprime ⇐⇒ M(A) π-complemented ⇐⇒ M(A) complemented.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 26 / 27
Finite dimensional ideals
TheoremEvery finite dimensional ideal of an m.s.p. algebra is complemented.
Lee-WongA prime associative algebra. A prime associative algebra A possessing a finitedimensional nonzero right ideal is finite dimensional and simple.
TheoremA nonzero m.p. algebraIf A has a nonzero finite dimensional ideal, then A is simple and finitedimensional.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 27 / 27
Finite dimensional ideals
Theorem
Every finite dimensional ideal of an m.s.p. algebra is complemented.
Lee-WongA prime associative algebra. A prime associative algebra A possessing a finitedimensional nonzero right ideal is finite dimensional and simple.
TheoremA nonzero m.p. algebraIf A has a nonzero finite dimensional ideal, then A is simple and finitedimensional.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 27 / 27
Finite dimensional ideals
TheoremEvery finite dimensional ideal of an m.s.p. algebra is complemented.
Lee-WongA prime associative algebra. A prime associative algebra A possessing a finitedimensional nonzero right ideal is finite dimensional and simple.
TheoremA nonzero m.p. algebraIf A has a nonzero finite dimensional ideal, then A is simple and finitedimensional.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 27 / 27
Finite dimensional ideals
TheoremEvery finite dimensional ideal of an m.s.p. algebra is complemented.
Lee-WongA prime associative algebra.
A prime associative algebra A possessing a finitedimensional nonzero right ideal is finite dimensional and simple.
TheoremA nonzero m.p. algebraIf A has a nonzero finite dimensional ideal, then A is simple and finitedimensional.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 27 / 27
Finite dimensional ideals
TheoremEvery finite dimensional ideal of an m.s.p. algebra is complemented.
Lee-WongA prime associative algebra. A prime associative algebra A possessing a finitedimensional nonzero right ideal
is finite dimensional and simple.
TheoremA nonzero m.p. algebraIf A has a nonzero finite dimensional ideal, then A is simple and finitedimensional.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 27 / 27
Finite dimensional ideals
TheoremEvery finite dimensional ideal of an m.s.p. algebra is complemented.
Lee-WongA prime associative algebra. A prime associative algebra A possessing a finitedimensional nonzero right ideal is finite dimensional and simple.
TheoremA nonzero m.p. algebraIf A has a nonzero finite dimensional ideal, then A is simple and finitedimensional.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 27 / 27
Finite dimensional ideals
TheoremEvery finite dimensional ideal of an m.s.p. algebra is complemented.
Lee-WongA prime associative algebra. A prime associative algebra A possessing a finitedimensional nonzero right ideal is finite dimensional and simple.
Theorem
A nonzero m.p. algebraIf A has a nonzero finite dimensional ideal, then A is simple and finitedimensional.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 27 / 27
Finite dimensional ideals
TheoremEvery finite dimensional ideal of an m.s.p. algebra is complemented.
Lee-WongA prime associative algebra. A prime associative algebra A possessing a finitedimensional nonzero right ideal is finite dimensional and simple.
TheoremA nonzero m.p. algebra
If A has a nonzero finite dimensional ideal, then A is simple and finitedimensional.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 27 / 27
Finite dimensional ideals
TheoremEvery finite dimensional ideal of an m.s.p. algebra is complemented.
Lee-WongA prime associative algebra. A prime associative algebra A possessing a finitedimensional nonzero right ideal is finite dimensional and simple.
TheoremA nonzero m.p. algebraIf A has a nonzero finite dimensional ideal,
then A is simple and finitedimensional.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 27 / 27
Finite dimensional ideals
TheoremEvery finite dimensional ideal of an m.s.p. algebra is complemented.
Lee-WongA prime associative algebra. A prime associative algebra A possessing a finitedimensional nonzero right ideal is finite dimensional and simple.
TheoremA nonzero m.p. algebraIf A has a nonzero finite dimensional ideal, then A is simple and finitedimensional.
Juan Carlos Cabello Píñar (Universidad de Granada) Multiplicatively semiprime algebras La Manga, 19-21 abril 2012 27 / 27