Kerry R. Sipe

Aug. 1, 2007

On the Non-Orientable Genus of Zero Divisor Graphs

Missouri State University

REU

Local Rings

Definition: A finite commutative ring R is local if it has a unique maximal ideal.

Definition: A maximal ideal of a ring R is an ideal M, not equal to R, such that there are no ideals “in between” M and R.

R

M

R

I

M is the maximal ideal of R. I is not the maximal ideal of R

J

Zero Divisors

Definition: An element

is a zero divisor of R if there is an element

such that

a R

b R

0.ab

When R is a local ring, the maximal ideal is exactly the set of zero divisors.

Zero Divisor Graph

Definition: The zero divisor graph of R, denoted

is the graph whose vertex set is the set of zero divisors of R and whose edge

set is

( )R

{{ , } ( ) | 0}.E a b Z R ab

16( ) {0,2,4,6,8,10,12,14}Z Z

Example: 16( ) Z

4

12

8

2

6

10

14

Non-Orientable Surfaces of Genus 1 and 2

Real Projective Plane Klein Bottle

A non-orientable surface cannot be embedded in 3-dimensional space without intersecting itself.

Genus

Definition: The non-orientable genus of a zero divisor graph is the smallest integer k such that the graph can be drawn on a surface of genus k without edges crossing.

Example: A planar graph has genus 0.

16( ) Z12

8

10

14

4

6

Example: A planar graph has genus 0.

2

Example: 3,3KCannot be drawn on the plane without intersecting itself.

Fundamental Polygons

Planar Genus 1

Genus 2

4K 3,3K

4,4K

Formulas for Finding the Genus of a Graph

,

( 2)( 2)( )

2m n

m nK

( 3)( 4)( )

6n

n nK

7( ) 3K

Formulas for determining the non-orientable genus of complete graphs and complete bipartite graphs:

with the exception :for

for

3n

, 2.m n

The Genus of Complete Graphs

Example: Complete graphs on n vertices:

( ) 0 1,2,3,4

( ) 1 5,6

( ) 2 7.

n

n

n

K for n

K for n

K for n

Example: Complete bipartite graphs:

,

3,3 3,4

4,4 3,5 3,6

( ) 0 1,2 .

( ) ( ) 1

( ) ( ) ( ) 2.

m nK for m and for all n

K K

K K K

Genus 1

Genus 2

My Game Plan:

Local Rings of order:

Non-local Rings

2

3

4

5

p

p

p

p

p

2

3

2

3

p q

p q

p q

p q

p q

Z

Z

F F

F F

F F

F

F

2

2

p p p

p q r

p q r

p q r

Z

F F F

F F F

F F F

F F

With two local factors:

With three local factors:

p p p p F F F FWith four local factors:

Theorem: Every finite commutative ring can be written as the product of local rings.

p is prime

Examples of Local Rings when p=2

8Z 16Z 32Z

2

6

414

8

4

12

2

6

10 22

2

10

6

26

18

30

16

24

8

28

12

20

4

8Z has p3 elements. 16Z has p4 elements. 32Z has p5 elements.

14

8ZM = {zero divisors} = (2) = {0, 2, 4, 6}

M2 = (4) = {0, 4}

M3 = (0) = {0}

4

2

6

The Maximal Ideal and the Zero Divisor Graph

The Maximal Ideal and the Zero Divisor Graph

16Z4

8

12

6

14

10

2

M = (2) = {0, 2,4,6,8,10,12,14}

M2 = (4) = {0, 4, 8, 12}

M3 = (8) = {0, 8}

M4 = (0) = {0}

The Maximal Ideal and the Zero Divisor Graph

32Z

14

22

2

10

6

26

18

30

16

24

8

28

12

20

4

M = (2) = {0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30}

M2 = (4) = {0, 4, 8, 12, 16, 20, 24, 28}

M3 = (8) = {0, 8, 16, 24}

M4 = (16) = {0, 16}

M5 = (0) = {0}

Collapsing the Graphs

M - M2 = (2) - (4) = {2, 6, 10, 14}

M2 - M3 = (4) – (8) = {4, 12}

M3 - M4 = (8) – (16) = {8}8

4

12

2

6

10

14

16Z

{ }a b R a b

( ) { 0}ann a r R ra

( ) ( )a b if ann a ann b

Making Vertex Sets From Equivalence Relations

Definition: The set of annihilators of a ring element is

.

a

Equivalence Relation:

In other words, two ring elements a and b are equivalent if they have the same annihilators.

8

4

12

2

6

10

14

16Z

[8] [2][4]

[2] { 2 } {2,6,10,14}b R b

An Example of an Equivalence Relation

(2)

(6)

(10)

(14) {0,8}

ann

ann

ann

ann

Collapsing the Graphs of Integer Rings

8Z

16Z

32Z

64Z[2]

[4]

[2]

[4]

[4][4]

[32]

[16]

[8]

[2]

[2]

[16]

[8]

[8]

What’s Next:

So far, we have been considering integer rings where M, M2, M3, …are each generated by one ring element.

[a3] [a][a2]

2 2

3 3

( )

( )

( )

.

.

.

{0}n

M a

M a

M a

M

What’s Next:What happens when M is generated by more than one element? For example:

2 2

3 3

4

( , )

( )

( )

{0}

M a b

M a

M a

M

[a3] [a]

[a2] [b]

44

2 2

3 3

4

[ ] (2 , )

(2, )

( )

( )

{0}

R X X X

M X

M X

M X

M

Z

The End

I’ll never forget my

time in Springfield.

[M2] [M5] [M3]

128Z

256Z

16

8

4

32