rings by rhemz and noemi

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INTRODUCTION

Investigation of algebraic structures and their properties is not limited only on

the study of one binary operation as for the groups. Another structure could be

defined in terms of two binary operations. The addition of one operation yields to

different properties and concepts which will be the concern of this report.

Rings

Definition: A ring is a non-empty set R together with two binary operations

+ and ., which we call addition and multiplication, defined on R such that the

following axioms are satisfied:

1. is an abelian group.

2. Multiplication is associative.

3. For all a, b, c R, the left distributive, a.(b+c) = (a.b) + (a.c) and the right

distributive law (a+b).c = (a.c) + (b.c) hold.

Examples(Rings):

1. Since the algebraic structures , , and satisfy

the properties for rings, these structures are rings.

2. Let F be the set of all functions f: R R . We know that is an abelian

group under the usual function addition,

(f+g)(x) = f(x) + g(x)

We define multiplication on F by

(fg)(x) = f(x)g(x)

That is, fg is the function whose value at x is f(x)g(x). Since the distributive

law holds for the structure then it is a ring.



3. Note that n Z is the cyclic subgroup of Z under addition consisting of all the

integer multiples of the integer n . Since (nr)(ns)= n(rs) , we see that n Z is

closed under multiplication. The associative and distributive laws which hold

in Z then assures that is a ring.

4. The cyclic group is a ring. Note that is an abelian group. It is

also observed that distributive laws and associative law for multiplication

under modulo n is satisfied.

6. (a). The set S = { a,b } with addition and multiplication defined by the tables

below is a ring.

+ a b

a a bb b a

a b

a a ab a b

(b). The set T = { a,b,c,d } with addition and multiplication defined by the

tables below is a ring.

+ a b c d

a a b c db b a d cc c d a bd d c b a

. a b c d

a a a a ab a b a bc a c a cd a d a d

7.Consider the set of positive rational numbers Q + . Clearly addition (

⊕ ) and multiplication ( ) defined by

a ⊕ b = a.b and a b = a+b for all a,b є Q +

where + and . are ordinary and multiplication on rational numbers, are

binary operations on Q + . Note that there are no inverses for each

element, and the distributive laws do not hold, so the set is not a ring.

Properties of Rings



The elementary properties of rings are analogous to those properties of Z

which do not depend upon either the commutative law of multiplication or

the existence of a multiplicative identity element. We call attention here to

some of these properties:

1. Every ring is an abelian additive group.

2. There exists a unique identity element z , (the zero of the ring)

3. Each element has a unique additive inverse, (the negative of that

element)

4. The Cancellation Law for addition holds.

5. for all a, b of the ring.

6.

7.

Subrings

Definition: Let R be a ring. A non-empty set S of the set R, which is itself a

ring with respect to the binary operations on R, is called a subring of R.

Note: When S is a subring of a ring R, it is evident that S is a subgroup of the

additive group R.

Proposition: Let S be a commutative ring, and let R be a nonempty subset of

S. Then R is a subring of S if and only if

(i) R is closed under addition and multiplication; and

(ii) if a belongs to R, then -a belongs to R.

Examples(Subring):

1. The trivial subring is a subring is a subring of every ring.



2. The Gaussian integers Z (i) = {a+bi/a,b ∈ Z } is a subring of the

complex numbers C.

3. In example 6(b), T 1= {a}, T 2= {a,b} are subrings of T. T 3 = {a,b,c} is

not a subring of T.

Definition: The subrings { z } and R itself of a ring R are called improper ;

other subrings, if any, of R are called proper .

Theorem 1 : Let R be a ring and S be a proper subset of the set R. Then S is a

subring of R if and only if

a. S is closed with respect to the ring operations.

b. for each a S , we have –a S.

Types of Rings

Commutative Rings

Definition: A ring for which multiplication is commutative is called

commutative ring.

Examples (Commutative Ring):

1. , , and are commutative rings.

2. 6(a) is a commutative ring and 6(b) is not a commutative ring.

Definition. Let R be a commutative ring with identity element 1. An

element a in R is said to be invertible if there exists an element b in R

such that ab = 1. The element a is also called a unit of R, and its

multiplicative inverse is usually denoted by a -1 .



Definition: An element e of a commutative ring R is said to be

idempotent if e 2 = e. An element a is said to be nilpotent if there

exists a positive integer n with a n = 0.

Proposition : Let R be a commutative ring with identity. Then the set R ×

of units of R is an abelian group under the multiplication of R.

A ring having a multiplicative identity element ( unit element

or unity ) is called a ring with identity element or ring with

unity .

Examples (Ring with Unity):

1. For the rings , , and the unity is 1.

2. For the ring on example 6(a), the unity is b and for example 6(b)

the ring has no unity.

Characteristic

Definition: Let R be a ring with zero element z and suppose that there exists

a positive integer n such that n . a = a+a+a+...+a = z for every a R . The

smallest such positive integer n is called the characteristic of R. If no such

integer exists, R is said to have characteristic zero.

Example(Characteristic): The ring Z n has characteristic n while ,

, and have characteristic zero.

Theorem 2: If R is a ring with unity 1, then R has characteristic n >0 if and

only if n is the smallest positive integer such that n.1 =0



Proof: By definition, if R has characteristic n >0, then n.a =0 for all a ∈ R, so

in particular n.1 =0. Conversely, suppose that n is a positive integer such that

n.1 =0. Then for any a ∈ R, we have

n . a = a+a+a+...+a = a (1+1+1+…1) = a(n .1) = a0= 0.

The theorem follows directly.

Divisors of Zero

Definition: Let R be a ring with zero element z . An element of R is called

a divisor of zero if there exists an element b of R such that or

.

Example: Find the divisors of zero in Z 12 defined by the equation x 2-5x +6 =

0.

Solution: The factorization x 2-5x +6 = (x-2)(x-3) is still valid if we think of x as

standing for any number in Z 12. But in Z 12 not only is 0a=a0 =0 for all a ∈ Z 12,

but also

(2)(6) = (6)(2) = (3)(4) = (4)(3)= (3)(8) = (8)(3) = (4)(6) =(6)(4)

= (4)(9)

= (9)(4) = (6)(6) = (6)(8) = (8)(6) = (6)(10) = (10)(6) =(8)(9) =

(9)(8)=0

Thus the equation has not only 2 and 3 as the solutions, but also 6 and 11,

for (6-2)(6-3) = (4)(3) = 0 and (11-2)(11-3) = (9)(8) = 0 in Z 12. The elements

2,3,4,6,8,9, and 10 are divisors of 0 in Z 12 .. Note that these numbers are

exactly the numbers in Z 12 that are not relatively prime to 12.

Homomorphisms and Isomorphisms



Definition: A homomorphism (isomorphism) of the additive group of a ring R

into (onto) the additive group of a ring R’ which also preserves the second

operation, multiplication, is called a homomorphism (isomorphism) of R into

(onto) R’ , that is

i. )()()( baba φ+φ=+φ and

ii. )()()( baab φφ=φ

Example(Homomorphism and Isomorphism):

1. The map : Z → Z n where (a) is the remainder of a modulo n is a

ring homomorphism for each positive integer n. We know that

)()()( baba φ+φ=+φ . To show that there is multiplicative property ,

write a=q 1n +r 1 and b=q 2n +r 2 according to the Division Algorithm.

Then ab = n (q 1 q 2+ r 1 q 2 + q 1 r 2 ) + r 1 r 2 . Thus φ (ab) is the remainder

of r 1 r 2 when divided by n. Since φ (a)= r 1 and φ (b)= r 2 , it shows that

)()( ba φφ is also the same remainder, so )()()( baab φφ=φ .

2. As abelian groups, +,

Z and +,2Z are isomorphic under the map

: Z → Z, with x x 2)( =φ for x ∈ Z . Here φ is not a ring isomorphism

for xy xy 2)( =φ , while .422)()( xy y x y x ==φφ

Theorem 1 : In any isomorphism of a ring R onto a ring R’ :

a. If z is the zero of R and z’ is the zero of R’ , we have z z’ .

b. If R R’ : a a’ , then –a -a’ .

c. If u is the unity of R and u’ is the unity of R’ , we have u u’ .

d. If R is a commutative ring, so also is R’ .

Ideals



Definition: Let R be a ring with zero element z. A subgroup S of R, having the

property r.x єS (x.rєS) for all xєS and rєR, is called a left (right) ideal in R.

Clearly, and R itself are both left and right ideals in R, they are

called improper left(right) ideals in R. All other left (right) ideals in R, if any,

are called proper.

Definition: A subgroup, ζ of R which is both a left and right ideal in R, that is,

for all x ∈ ζ and r ∈ R both r.x є ζ and x.r є ζ, is called an ideal(invariant

subring) in R.

Note: A ring having no proper ideals is called a simple ring.

Example(Ideals):

1. n Z is an ideal in the ring Z since we know it is a subring, and s(nm) =

(nm)s= n(ms) ∈ n Z for all s ∈ Z.

2. Let F be the ring of all functions mapping R onto R , and let C be the

subring of F consisting of all the constant functions in F. C is not an

ideal since it is not true that the product of a constant function with

every function is again a constant function. For example, the product

of sin x and 2 is the function 2 sinx. .

Quotient Rings(Factor Rings)

Definition: Let N be an ideal of a ring R. Then the additive cosets of N form a

ring R/N with the binary operations defined by

(a+N) +(b+N) = (a+b) + N

and (a+N)(b+N) = (ab) + N

The ring R/N is called a factor ring(or quotient ring) of R modulo N.



Example( Factor Ring): Consider the ideal N = {3r: r ∈ Z } of the ring Z and

the quotient group Z N= { N, 1+N, 2+N}. It is clear that the elements of Z N

are the residue classes of Z 3, and thus constitute a ring with respect to

addition and multiplication modulo 3.

Theorem2: If N is an ideal in a ring, the quotient group R/N is a ring with

respect to addition and multiplication of cosets (residue classes).

From the definition of addition and multiplication of residue classes, it

follows that:

i. The mapping R → R/N : a + a +N is a homomorphism of R onto

R/N.

ii. N is the zero element of the ring R/N

iii. If R is a commutative ring, so also is R/N

iv. If R has a unity element u , so also has R/N, namely u + N.

v. Ig R is without divisors of zero, R/N may or may not have divisors

of zero. For, while (a+N)(b+N) = (ab) + N= N indicates a.b ∈

N, it does not necessarily imply either a ∈ N or b ∈ N.

Euclidean Ring

Definition: Let R be a commutative ring having the property that to each x

∈ R a non-negative integer )( x θ can be assigned such that

i. )( x θ = 0 if and only if x= z, the zero element of R

ii. ≥θ )( xy )( x θ when x.y ≠ z .

iii. For every x ∈ R and y ≠ z ∈ R

x= y.q +r q,r ∈ R, 0 ≤ )( r θ < )( y θ



Example(Euclidean Ring): Z is a Euclidean ring. This follows easily by using

)( x θ = x for every x ∈ R.

INSIGHTS:

1. The addition of one binary operation defines another algebraic

structure- the rings. The study of this structure needs wider analysis

and perspective.

2. Rings are extension of groups. The basis of investigation of rings is the

properties associated with groups under addition and multiplication as

binary operations. Since it is based on the groups, some concepts that

are under rings are patterned from groups such as the subrings,

isomorphisms on rings and cosets on rings.



References:

Ayres, Frank and Lloyd R. Jaisingh. Schaum’s Outlines of Theory and Problems

of Abstract Algebra, 2 nd Ed. USA: Mc Graw Hill Companies, Inc., 2004.

Fraleigh, John B. A First Course in Abstract Algebra, Sixth Ed. USA: Addison-

Wesley Publishing Company, 2000.



Exercises (RINGS):

1. (a) Show that { }Z∈= x x S :2 with addition and multiplication as defined

on Z is a ring while { }Z∈+= x x T :12 is not.

2. Verify that { }gf ed cbaS ,,,,,,= with addition and multiplication

defined by

is a ring. (a) Is it commutative ring? (b) Find the subrings of the ring. (c) What

is the unity? (d) Find the idempotent and nilpotent elements. (e)What is the

characteristic? (f) Does it have divisors of zero?

+ a b c d e f g

a a b c d e f g

b b c d e f g a

c c d e f g a b

d d e f g a b c

e e f g a b c d

f f g a b c d e

g g a b c d e f

◦ a b c d e f g

a a a a a a a a

b a b c d e f g

c a c e g b d f d a d g c f b e

e a e b f c g d

f a f d b g e c

g a g f e d c b

rings by rhemz and noemi

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