Introduction/BackgroundOur Methods

Results and ConclusionsReferences

Constructing Zero Divisor Graphs

Alaina Wickboldt, Louisiana State UniversityAlonza Terry, Xavier University of LouisianaCarlos Lopez, Mississippi State University

SMILE 2011

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

Outline

Introduction/BackgroundRing TheoryModular ArithmeticZero Divisor Graphs

Our MethodsDrawing by handImplementing Computer Coding

Results and ConclusionsResultsConclusion

References

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

Joint work with:• Dr. Sandra Spiro↵, University of Mississippi

• Dr. Dave Chapman, Louisiana State University

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

THE PROJECT

To study the various definitions of zero divisor graphs associated torings, namely those by I. Beck [B] and D. Anderson & P.Livingston [AL].

• Construct the [AL] graph �(R) for R = (Zn) up toorder n, for n 100.• Observe behavior and patterns of each graph.

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

THE PROJECT

To study the various definitions of zero divisor graphs associated torings, namely those by I. Beck [B] and D. Anderson & P.Livingston [AL].

• Construct the [AL] graph �(R) for R = (Zn) up toorder n, for n 100.• Observe behavior and patterns of each graph.

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

Ring TheoryModular ArithmeticZero Divisor Graphs

Relations Between Rings and Graphs

Ring

Z/6Z

Graph

2

<<<<

<<<<

⇥⇥⇥⇥⇥⇥⇥⇥

1

3

<<<<

<<<<

0

⇥⇥⇥⇥⇥⇥⇥⇥

<<<<

<<<<

⇥⇥⇥⇥⇥⇥⇥⇥

<<<<<<<<

G (Z6)

4 5

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

Ring TheoryModular ArithmeticZero Divisor Graphs

A ring R is a set together with two binary operations + and ·(called addition and multiplication) satisfying the following axioms:

A. R is an Abelian group under +; i.e.,

B. For any a, b in R , ab is in R . (closure of multiplication)

C. For any a, b, c in R , a(bc) = (ab)c . (associativity of

multiplication)

D. For any a, b, c in R , a(b + c) = ab + ac and(a+ b)c = ac + bc . (distributive property)

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

Ring TheoryModular ArithmeticZero Divisor Graphs

A ring R is a set together with two binary operations + and ·(called addition and multiplication) satisfying the following axioms:

A. R is an Abelian group under +; i.e.,

B. For any a, b in R , ab is in R . (closure of multiplication)

C. For any a, b, c in R , a(bc) = (ab)c . (associativity of

multiplication)

D. For any a, b, c in R , a(b + c) = ab + ac and(a+ b)c = ac + bc . (distributive property)

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

Ring TheoryModular ArithmeticZero Divisor Graphs

Example

Let R be the integers with arithmetic modulo 6. The elements ofR are {0, 1, 2, 3, 4, 5}, and the operations of addition andmultiplication are detailed in the tables below:

+ 0 1 2 3 4 50 0 1 2 3 4 51 1 2 3 4 5 02 2 3 4 5 0 13 3 4 5 0 1 24 4 5 0 1 2 35 5 0 1 2 3 4

* 0 1 2 3 4 50 0 0 0 0 0 01 0 1 2 3 4 52 0 2 4 0 2 43 0 3 0 3 0 34 0 4 2 0 4 25 0 5 4 3 2 1

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

Ring TheoryModular ArithmeticZero Divisor Graphs

Example

Let R be the integers with arithmetic modulo 6. The elements ofR are {0, 1, 2, 3, 4, 5}, and the operations of addition andmultiplication are detailed in the tables below:

+ 0 1 2 3 4 50 0 1 2 3 4 51 1 2 3 4 5 02 2 3 4 5 0 13 3 4 5 0 1 24 4 5 0 1 2 35 5 0 1 2 3 4

* 0 1 2 3 4 50 0 0 0 0 0 01 0 1 2 3 4 52 0 2 4 0 2 43 0 3 0 3 0 34 0 4 2 0 4 25 0 5 4 3 2 1

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

Ring TheoryModular ArithmeticZero Divisor Graphs

Zero Divisor Graph: The zero divisor graph of a commutative ringR with identity is a simple graph whose set of vertices consists ofall elements of the ring, with an edge defined between a and b ifand only if ab = 0.

Unit: A unit is a non-zero element u in a commutative ring withidentity such that there is another non-zero element v of the ringsatisfying uv = 1.

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

Ring TheoryModular ArithmeticZero Divisor Graphs

Zero Divisor Graph: The zero divisor graph of a commutative ringR with identity is a simple graph whose set of vertices consists ofall elements of the ring, with an edge defined between a and b ifand only if ab = 0.

Unit: A unit is a non-zero element u in a commutative ring withidentity such that there is another non-zero element v of the ringsatisfying uv = 1.

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

Ring TheoryModular ArithmeticZero Divisor Graphs

Complete Graph: A graph is complete if every vertex in the graphis adjacent to every other vertex.

Complete Bipartite Graph: A complete bipartite graph is a graphwhose vertex set can be partitioned into 2 disjoint subset ui and vj

such each ui is adjacent to every vj , but no two ui ’s are adjacentand no two vj ’s are adjacent.

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

Ring TheoryModular ArithmeticZero Divisor Graphs

Complete Graph: A graph is complete if every vertex in the graphis adjacent to every other vertex.

Complete Bipartite Graph: A complete bipartite graph is a graphwhose vertex set can be partitioned into 2 disjoint subset ui and vj

such each ui is adjacent to every vj , but no two ui ’s are adjacentand no two vj ’s are adjacent.

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

Ring TheoryModular ArithmeticZero Divisor Graphs

Definition: [I. Beck, 1988] The (original) zero divisor graph of aring R is a simple graph whose set of vertices consists of allelements of the ring, with an edge defined between a and b if andonly if ab = 0. It will be denoted by G (R).

Example R = Z6 V (R) = Z6 = {0, 1, 2, 3, 4, 5}2

<<<<

<<<<

⇥⇥⇥⇥⇥⇥⇥⇥

1

3

<<<<

<<<<

0

⇥⇥⇥⇥⇥⇥⇥⇥

<<<<

<<<<

⇥⇥⇥⇥⇥⇥⇥⇥

<<<<<<<<

G (Z6)

4 5

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

Ring TheoryModular ArithmeticZero Divisor Graphs

Definition: [I. Beck, 1988] The (original) zero divisor graph of aring R is a simple graph whose set of vertices consists of allelements of the ring, with an edge defined between a and b if andonly if ab = 0. It will be denoted by G (R).

Example R = Z6 V (R) = Z6 = {0, 1, 2, 3, 4, 5}2

<<<<

<<<<

⇥⇥⇥⇥⇥⇥⇥⇥

1

3

<<<<

<<<<

0

⇥⇥⇥⇥⇥⇥⇥⇥

<<<<

<<<<

⇥⇥⇥⇥⇥⇥⇥⇥

<<<<<<<<

G (Z6)

4 5

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

Ring TheoryModular ArithmeticZero Divisor Graphs

Definition: [D. F. Anderson & P. Livingston, 1999] The zerodivisor graph of a ring R is a simple graph whose set of verticesconsists of all (non-zero) zero divisors, with an edge definedbetween a and b if and only if ab = 0. It will be denoted by �(R).

Example R = Z6 V (R) = Z

⇤(Z6) = {2, 3, 4}

2

⇥⇥⇥⇥⇥⇥⇥⇥

1

3

<<<<

<<<<

0 �(Z6)

4 5

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

Ring TheoryModular ArithmeticZero Divisor Graphs

Definition: [D. F. Anderson & P. Livingston, 1999] The zerodivisor graph of a ring R is a simple graph whose set of verticesconsists of all (non-zero) zero divisors, with an edge definedbetween a and b if and only if ab = 0. It will be denoted by �(R).

Example R = Z6 V (R) = Z

⇤(Z6) = {2, 3, 4}

2

⇥⇥⇥⇥⇥⇥⇥⇥

1

3

<<<<

<<<<

0 �(Z6)

4 5

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

Zero Divisor GraphsImplementing Computer Coding

Methods

• Our initial approach to this task was to draw as many graphs aswe could by hand.

• After many tedious drawings we realized that once you got intolarger numbers, the graphs became exceedingly complex to justdraw by hand.

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

Zero Divisor GraphsImplementing Computer Coding

• To assist us in understanding the properties of the graphs wecreated better representations of the graphs.

• We began to focus on the properties of each number and on it’srelation with it’s individual graph.

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

Zero Divisor GraphsImplementing Computer Coding

• To assist us in understanding the properties of the graphs wecreated better representations of the graphs.

• We began to focus on the properties of each number and on it’srelation with it’s individual graph.

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

Zero Divisor GraphsImplementing Computer Coding

• Even though we made the graphs a little more presentable itstill became a little tedious to draw the graphs of Z mod n forlarger n values.

• Mathematical computer programs became our best friend.

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

Zero Divisor GraphsImplementing Computer Coding

• Even though we made the graphs a little more presentable itstill became a little tedious to draw the graphs of Z mod n forlarger n values.

• Mathematical computer programs became our best friend.

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

Zero Divisor GraphsImplementing Computer Coding

MATHEMATICA!!

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

ResultsConclusion

Results

• Prime Factorization.• Categories.

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

ResultsConclusion

One Prime Factors

• Prime numbers have no Anderson and Livingston Graph.

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

ResultsConclusion

Two Prime Factors

Represented by p and q

• Distinct Case

Represented by p

2

• Non-Distinct Case

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

ResultsConclusion

Two Prime Factors

Represented by p and q

• Distinct Case

Represented by p

2

• Non-Distinct Case

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

ResultsConclusion

Distinct Case:

Figure: �(Z33)

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

ResultsConclusion

Non-Distinct Case:

Figure: �(Z25)

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

ResultsConclusion

Three Prime Factors

This can be represented three di↵erent ways.

• p

3

• p

2q

• pqr

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

ResultsConclusion

First Case : p3

Figure: �(Z27)

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

ResultsConclusion

Second Case: p2q

Figure: �(Z18)

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

ResultsConclusion

Third Case: pqr

Figure: �(Z30)

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

ResultsConclusion

Four Prime Factors

This can be represented four di↵erent ways.

• p

4

• p

3q

• p

2qr

• p

2q

2

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

ResultsConclusion

First Case: p4

Figure: �(Z81)

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

ResultsConclusion

Second Case: p3q

Figure: �(Z24)

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

ResultsConclusion

Third Case: p2qr

Figure: �(Z60)

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

ResultsConclusion

Fourth Case: p2q2

Figure: �(Z36)

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

ResultsConclusion

Five Prime Factors

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

ResultsConclusion

Example: p3q2

Figure: �(Z72)

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

ResultsConclusion

Six Prime Factors

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

ResultsConclusion

Example: p5q

Figure: �(Z96)

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

ResultsConclusion

Just for some fun....

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

ResultsConclusion

Example

Figure: �(Z6561)

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

ResultsConclusion

THE CONCLUSION

Based on our study of over 100 zero divisor graphs, we were ableto conclude that the complexity of the graph is dependent on thecomplexity of the prime factorization of the number we are doingmodular arithmetic with.

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs

Introduction/BackgroundOur Methods

Results and ConclusionsReferences

References

1. D. Anderson, P. Livingston, The zero-divisor graph of acommutative ring, J. Algebra, 217 (1999)

2. Dr. Spiro↵, Sandra. University of Mississippi.

3. Dr. Chapman, Dave. Louisiana State University.

4. B. Kelly and E. Wilson, Investigating the algebraic structureembedded in zero-divisor graphs, to appear, American J. ofUndergraduate Research.

A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs