Classification of Zero Divisor Graphs of a Commutative Ring With Degree Equal 7 and 8

123 Classification of Zero Divisor Graphs of a Commutative Ring With Degree Equal 7 and 8 Nazar H. Shuker nazarh_2013@yahoo.com Husam Q. Mohammad husam_alsabawi@yahoo.com College of Computer Sciences and Mathematics University of Mosul, Mosul, Iraq Received on: 15/05/2012 Accepted on: 18/09/2012 ABSTRACT In 2005 J. T Wang investigated the zero divisor graphs of degrees 5 and 6. In this paper, we consider the zero divisor graphs of a commutative rings of degrees 7 and 8.


Introduction
The concept of zero divisor graph of a commutative ring was introduced by Beck in [3]. He let all the elements of the ring be vertices of the graph. In [1] Anderson and Livingston introduced and studied the zero divisor graph whose vertices are the non-zero zero divisors.
Throughout this paper, all rings are assumed to be commutative rings with identity, and Z(R) be the set of zero divisors. We associate a simple graph Γ(R) to a ring R with vertices Z(R) * = Z(R)-{0}, the set of all non-zero zero divisors of R. For all distinct x,y Z(R) * , the vertices x and y are adjacent if and only if xy=0. (R,m) and |S| will stand respectively for the local ring with maximal ideal m and cardinal numbers of a set S.
In [1] Anderson and Livingston proved that for any commutative ring R Γ(R) is connected.
In 2005 J. T Wang [5] investigated the zero divisor graphs of degrees 5 and 6. In this paper, we extend this results to consider the zero divisor graphs of commutative rings of degrees 7 and 8.
The main result when |Z(R) * |=7 is given in Theorem 2.7,while when |Z(R) * |=8 the main result is given in Theorem 3.4. We also extend Wang's result concerning local rings (Theorem 2.2)

Rings with |Z(R) * |=7
It is known that if R is a ring then Γ(R) is connected. In this section, we find all possible graphs of Γ(R) with Γ(R)=7.
Recall that if R is finite ring, then every element of R is either a unit or a zero divisor [2]. In [5] Wang proved the following result.
For any non-zero-divisor (a,b) in R(1)(2)xR3, we have the following cases: 1-If a is non-zero divisor of R(1)(2), then a must be a unit element. If b is a zero divisor of R3, then there are (|R(1)(2)|-|Z(R(1)(2))|)x|m3| elements of this type. 2-If a is a non-zero zero divisor of R(1)(2) and b any element in R3, then there are (|Z(R(1)(2))|-1)x|R3| elements of this type. 3-If a=0, and b is a non-zero element in R3, then there are 1x(|R3|-1). Now, we sum up these three types of elements; there are as follows: As a direct consequence to Theorem 2.2, we obtain the following:

Corollary 2.4 :
If R finite and RR1xR2xR3, then |Z(R)*|≥13 for some local rings Ri but not field.

Lemma 2.5 :
If R is a ring with |Z(R)| * =7, then is either R local ring or R is isomorphic to a product of two local rings.

Rings with |Z(R) * |=8
The main aim of this section is to find all possible zero divisor graphs of 8 vertices and rings which correspond to them.
We shall start this section with following lemmas which play a central role in the sequel.

Lemma 3.1 :
Let R be a ring with |Z(R)| * =8, then R is local or R is isomorphic to a product of two local rings.

Lemma 3.2 :
Let R be a ring which is not local and |Z(R)| * =8, then RF1xF2, where F1 and F2 are fields Proof: Since R not local, then by Lemma 3. Let R be a ring which is not local and |Z(R) * | =8, then the following graph can be realized as Γ(R). By Lemma 3.3, then R Z2xF8 or Z3xZ7 or Z5xZ5. In Figure (1), can be realized as Γ(Z2xF8 ). Figure (2) , can be realized as Γ(Z3xZ7). Figure (3) , can be realized as Γ(Z5xZ5). ■