The Graph of Annihilating Ideals

Let R be a commutative ring with identity and AG(R) be the set of ideals with non-zero annihilators. The annihilating ideal graph AG(R) is a graph of vertex set AG (R)\{(0)} and two distinct ideal vertices I and J are adjacent if and only if I J = (0). In this paper , we establish a new fundamental properties of AG(R) as well as its connection with Γ(R).


Introduction:
Let R be a commutative ring with identity , and let Z(R) be its set of zero divisors. We associate a simple graph Γ(R) to R with vertices Z * (R)=Z(R)\{(0)} , the set of all non-zero zero divisors of R , and for distinct , ∈Z * (R) , the vertices x and y are adjacent if and only if xy=0. Thus , Γ(R) is empty graph iff R is an integral domain.
Beck introduced the concept of zero divisor graph of a commutative ring in [4]. In the recent years zero divisor graph have been extensively studied by many authors in [1,2,3,8].
An ideal I of R is said to be annihilating ideal if there exists a non-trivial ideal J of R such that I J=(0). Let AG(R) be the set of annihilating ideals of R. The annihilating ideal graph AG(R) is a graph with vertex set AG * (R)=AG(R)\{(0)} such that there is an edge between vertices I and J if and only if I≠ and I J = (0). The idea of annihilating ideal graph was introduced by Behboodi and Rakeei in [5] and [6].
In the present paper , we investigate the annihilating ideal graph AG(R). We establish a new of its basic properties and its relation of Γ( ).
Recall that: 1. R is called reduced if R has no non-zero nilpotent element. 2.The distance d(u,v) between two vertices u and v of a connected graph Γ is the minimum of the lengths of the u-v paths of Γ [7]. 3. The degree of the vertex a in the graph Γ is the number of edges of Γ incident with a [7]. 4. The graph Г is called a plane graph if it can be drawn in the plane with their edges crossing. A graph which is an isomorphic to a plane graph is called a planer graph [7]. 5. A graph Г is bipartite graph , if it is possible to partition the vertex set of Г into two subsets V1 and V2 such that every element of edges of Г joins a vertex of V1 to a vertex of V2. A complete bipartite graph with partite sets V1 and V2 where , | 1 |=m and | 2 |=n , is then denoted by Km,n [7].

Annihilating ideal graph:
In this section , we consider annihilating ideal graph, we give some of its basic properties and provide some examples.

Definition2.1[5]:
Let R be a ring and let I and J are distinct non-trivial ideals of R.Then , I and J are adjacent ideal vertices in AG(R) if I J=(0).
From now on , we shall use the symbol I-J to denote for two adjacent ideal vertices I and J. We start this section with the following example .
Example1: Let Z24 be the ring of integers modulo 24. The graph AG(Z24) can be drawn as follows: The following result is an easy consequence of definition of 2.1.

Lemma2.2:
If I and J are non-trivial ideals of R such that ∩ = (0) , then I-J is an edge of AG(R) and ∪ ⊆Z(R).
The converse of Lemma2.2 is not true in general, as the following example shows.
The next result considers the number of minimal ideals of R .

Theorem2.7: If
( ) is a planar graph , then R has at most four minimal ideals. Proof: Suppose that R has five minimal ideals say M1 , M2 , M3 , M4 and M5. By Proposition2.6 , any two of M1 , M2 , M3 , M4 and M5 are adjacent. This means that ( ) contains the complete graph K5. This is contradiction that ( ) is a planar graph (See the Kuratowsky Theorem in [7]). Therefore, R has at most four minimal ideals. ∎ Example5: Let Z16 be the ring of integers modulo 16.
Clearly, the graph AG(Z16) is a planar graph and the only minimal ideal of Z16 is (8).

The graphs ( ) and
( ) In this section, we consider the relationship between ( ) and (R). It is natural to ask whether ( ) and (R) are isomorphic , the answer is negative , as the following example shows.
Example6: Let Z12 be the ring of integer modulo 12. Then, the number of vertices of Γ(Z 12 ) is 7 , while the number of vertices of (Z 12 ) is 4. Obviously, Γ(Z 12 ) and (Z 12 ) are not isomorphic.