kerry r. sipe aug. 1, 2007
DESCRIPTION
On the Non-Orientable Genus of Zero Divisor Graphs. Kerry R. Sipe Aug. 1, 2007. Missouri State University REU. Local Rings. 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. - PowerPoint PPT PresentationTRANSCRIPT
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