Zero-Divisor Graph with Respect to an Ideal^{*}

H. R. Maimani, M. R. Pournaki, S. Yassemi

Abstract Let R be a commutative ring with nonzero identity and let I be an ideal of R. The zero-divisor graph of R with respect to I, denoted by Γ_{I}(R), is the graph whose vertices are the set {x ∈ R \ I| xy ∈ I for some y ∈ R \ I} with distinct vertices x and y adjacent if and only if xy ∈ I. In the case I = 0, Γ_{0}(R), denoted by Γ(R), is the zero-divisor graph which has well known results in the literature. In this article we explore the relationship between Γ_{I}(R) ≅ Γ_{J}(S) and Γ(R/I) ≅ Γ(S/J). We also discuss when Γ_{I}(R) is bipartite. Finally we give some results on the subgraphs and the parameters of Γ_{I}(R).

Keywords: Zero-divisor graph, r-Partite graph, Clique number, Girth. 2000 Mathematics Subject Classification: 05C75, 13A15.

1 Introduction and Preliminaries

Let R be a commutative ring with nonzero identity, and let Z(R) be its set of zero divisors. The zero-divisor graph, Γ(R), is the graph with vertices Z(R)^{∗} = Z(R)\{0}, the set of nonzero zero divisors of R, and for distinct x, y ∈ Z(R)^{∗}, the vertices x and y are adjacent if and only if xy = 0. In [7] Beck introduced the concept of a zero-divisor graph of a commutative ring. However, he lets all elements of R be vertices of the graph and his work was mostly concerned with coloring of rings. We adopt the approach used by D. F. Anderson and P. S. Livingston in [6] and consider only nonzero zero divisors as vertices of the graph. The zero-divisor graph of a commutative ring has been studied extensively by several authors, e.g., [7, 6, 4, 10, 5, 1, 2].

In [11] Redmond introduced the definition of the zero-divisor graph with respect to an ideal. Let I be an ideal of R. The zero-divisor graph of R with respect to I, denoted by Γ_{I}(R), is the graph whose vertices are the set {x ∈ R \ I| xy ∈ I for some y ∈ R \ I} with distinct vertices x and y adjacent if and only if xy ∈ I. Thus if I = 0 then Γ_{I}(R) = Γ(R), and I is a nonzero prime ideal of R if and only if Γ_{I}(R) = ∅. In [11] Redmond explored the relationship between Γ_{I}(R) and Γ(R/I). He gave an example of rings R and S and ideals I ⊴ R and J ⊴ S, where Γ(R/I) ≅ Γ(S/J) but Γ_{I}(R) ≇ Γ_{J}(S). Among other things, he showed that for an ideal I of R, Γ_{I}(R) contains |I| disjoint subgraphs isomorphic to Γ(R/I). In section 2, we show that for finite ideals I and J of R and S, respectively, for which I = \sqrt{I} and J = \sqrt{J}, if Γ_{I}(R) ≅ Γ_{J}(S), then Γ(R/I) ≅ Γ(S/J). Also we will show that the converse of this result holds if |I| = |J| (see Theorem 2.2).

*The research of the first author was in part supported by grant no. 83050119 from IPM.

The research of the second author was in part supported by a grant from IPM.

The research of the third author was in part supported by grant no. 83130214 from IPM.

For a graph G, the vertices set of G is denoted by V (G). The degree of a vertex v in G is the number of edges of G incident with v. We denote by δ(G) the minimum degree of vertices of G. For any nonnegative integer r, the graph G is called r-regular if the degree of each vertex is equal to r. The girth of G is the length of a shortest cycle in G and is denoted by gr(G). If G has no cycles, we define the girth of G to be infinite. An r-partite graph is one whose vertex set can be partitioned into r subsets so that no edge has both ends in any one subset. A complete r-partite graph is one in which each vertex is jointed to every vertex that is not in the same subset. The complete bipartite (i.e., 2-partite) graph with part sizes m and n is denoted by K_{m,n}. A graph in which each pair of distinct vertices is jointed by an edge is called a complete graph. We use K_{n} for the complete graph with n vertices. In section 3, we show that Γ_{I}(R) is a complete bipartite graph provided I = 𝔭_{1} ∩ 𝔭_{2} ≠ 0 for prime ideals 𝔭_{1} and 𝔭_{2} of R (see Theorem 3.1).

A clique of a graph is a maximal complete subgraph and the number of graph vertices in the largest clique of graph G, denoted by ω(G), is called the clique number of G. In section 4, we show that if I is an ideal of R such that I = ⋂_{1≤i≤n} 𝔭_{i} and for each 1 ≤ j ≤ n, I ≠ ⋂_{1≤i≤n, i≠j} 𝔭_{i} where 𝔭_{i}’s are prime ideals of R, then ω(Γ_{I}(R)) = n (see Theorem 4.2).

In this article the notations of graph theory are from [8], and the notations of commutative rings are from [9].

2 Some Basic Properties of Zero-Divisor Graphs

One of the main questions in the study of zero-divisor graphs is as follows: Let R and S be two commutative rings. If Γ(R) ≅ Γ(S), then do we have R ≅ S? Some well known results on this question are as follows:

(i) If R and S are two finite reduced rings which are not fields, and Γ(R) ≅ Γ(S), then R ≅ S (see [4, Theorem 4.1]).

(ii) If R is a finite reduced ring which is not isomorphic to ℤ_{2} ⊕ ℤ_{2} or ℤ_{6}, and S is a ring which is not a local integral domain, and Γ(R) ≅ Γ(S), then R ≅ S (see [2, Theorem 5]).

(iii) If R = ∏_{i∈I} F_{i} and S = ∏_{j∈J} G_{j}, where F_{i}’s are finite fields and G_{j}’s are integral domains, and Γ(R) ≅ Γ(S), then R ≅ S (see [5, Theorem 2.1]).

Now let I be an ideal of R and J be an ideal of S. It is natural to ask the following question. If Γ_{I}(R) ≅ Γ_{J}(S), then do we have R/I ≅ S/J? The main purpose of this section is to focus on this question.

A subgraph H of G is called a spanning subgraph when V (G) = V (H). A 1regular spanning subgraph H of G is called a 1-factor or a perfect matching of G. A graph G is 1-factorable if the edges of G are partitioned into 1-factors of G. Every r-regular bipartite graph is 1-factorable (cf. [8, p. 192]). If the edges of G are partitioned into subgraphs H_{1}, …, H_{n}, then we write G ≅ H_{1} ⊕ … ⊕ H_{n}, and if H_{i} ≅ H_{j} for all 1 ≤ i, j ≤ n, then we write G ≅ nH, where H ≅ H_{i}.

Theorem 2.1 Let I be a finite ideal of R such that I = \sqrt{I}. Then Γ_{I}(R) ≅ |I|^{2}Γ(R/I).

Proof. Let e be the edge of Γ(R/I) between the vertices a and b. Since every element of coset a + I is adjacent to every element of coset b + I, it is easy to see that there exists a subgraph of Γ_{I}(R), denoted by H^{(e)}, which is isomorphic to complete bipartite graph K_{|I|,|I|}. On the other hand, by [8, p. 192], we have K_{|I|,|I|} ≅ M_{1}^{(e)} ⊕ … ⊕ M_{|I|}^{(e)}, where each of M_{i}^{(e)} is a perfect matching of K_{|I|,|I|}. Now consider K_{i} := ⊕_{e∈E(Γ(R/I))}M_{i}^{(e)} which is a subgraph of Γ_{I}(R). Since I = \sqrt{I}, Γ_{I}(R) ≅ K_{1} ⊕ … ⊕ K_{|I|}. Now the assertion follows from the fact that each K_{i} is partitioned into |I| edge-disjoint subgraphs, where each of them is isomorphic to Γ(R/I). □

Let S be a nonempty set of vertices of a graph G. The subgraph induced by S is the maximal subgraph of G with vertex set S, and is denoted by ⟨S⟩, that is, ⟨S⟩ contains precisely those edges of G joining two vertices in S.

Theorem 2.2 Let I be a finite ideal of R and let J be a finite ideal of S such that I = \sqrt{I} and J = \sqrt{J}. Then the following hold:

(a) If |I| = |J| and Γ(R/I) ≅ Γ(S/J), then Γ_{I}(R) ≅ Γ_{J}(S).

(b) If Γ_{I}(R) ≅ Γ_{J}(S), then Γ(R/I) ≅ Γ(S/J).

Proof. Part (a) is an easy consequence of Theorem 2.1. For proving part (b), let φ : Γ_{I}(R) ⟶ Γ_{J}(S) be an isomorphism. Now consider K ⊆ R to be a set of distinct representatives of the vertices of Γ(R/I). Clearly, the subgraph induced by K is isomorphic to Γ(R/I). Now consider the restriction of φ to K. Suppose that φ(K) = K′ and ⟨K′⟩ = H. Now, if a, b ∈ V (K′), then a + J ≠ b + J; otherwise, a^{2} ∈ J = \sqrt{J}, and hence a ∈ J, which is a contradiction. Hence, K′ is a distinct representation of the vertices of Γ(S/J), and hence ⟨K′⟩ = H ≅ Γ_{J}(S). Therefore, φ induced an isomorphism from Γ(R/I) to Γ(S/J). □

Note that in Theorem 2.2 (a), the condition “|I| = |J|” is not superficial, as the following example shows.

Example 2.3 Let R = ℤ_{3} ⊕ ℤ_{3} ⊕ ℤ_{3} ⊕ ℤ_{3} and S = ℤ_{3} ⊕ ℤ_{3} ⊕ ℤ_{3} ⊕ ℤ_{3} ⊕ ℤ_{3}, and consider I = ℤ_{3} ⊕ ℤ_{3} and J = ℤ_{3} ⊕ ℤ_{3} ⊕ ℤ_{3}. Hence, Γ(R/I) ≅ Γ(S/J). But by computing the number of edges in each graph we have Γ_{I}(R) ≇ Γ_{J}(S).

The conditions “I = \sqrt{I}” and “J = \sqrt{J}” on ideals I and J are also necessary in Theorem 2.2 (see [11, Remark 2.3]).

Theorem 2.4 Let I be a nonzero ideal of R and a ∈ Γ_{I}(R), adjacent to every vertex of Γ_{I}(R). Then (I : a) is a maximal element of the set {(I : x)| x ∈ R}. Moreover, (I : a) is a prime ideal.

Proof. Let V = V (Γ_{I}(R)). Choose 0 ≠ x ∈ I. It is easy to see that a ≠ a + x ∈ Γ_{I}(R). Thus a(a + x) ∈ I and hence a^{2} ∈ I. Therefore, V ∪ I = (I : a), and so for any x ∈ R, (I : x) is contained in V ∪ I = (I : a). Thus the first assertion holds.

Now, we prove that (I : a) is a prime ideal. Let xy ∈ (I : a) and x, y ∉ (I : a). Therefore, xya ∈ I. If ya ∉ I, then x ∈ (I : ya). We know that (I : a) ⊆ (I : ya), and therefore, (I : a) = (I : ya). Hence, x ∈ (I : a), which is a contradiction. □

Theorem 2.5 Let I be an ideal of R and let S be a clique in Γ_{I}(R) such that x^{2} = 0 for all x ∈ S. Then S ∪ I is an ideal of R.

Proof. Suppose that x, y ∈ S ∪ I. Consider the following three cases.

Case 1: If x, y ∈ I, then x − y ∈ S ∪ I.

Case 2: If x, y ∈ S with x − y ∉ I, then for all c ∈ S, c(x − y) ∈ I and hence S ∪ {x − y} is a clique. Now, since S is a clique, x − y ∈ S.

Case 3: If x ∈ I and y ∈ S, then x−y ∉ I, and hence for any c ∈ S, c(x−y) ∈ I. Therefore, x − y ∈ S.

Now, let x ∈ S ∪ I and r ∈ R. Suppose that r, x ∉ I. If rx ∈ I, then rx ∈ S ∪ I.

If rx ∉ I, since for any c ∈ S, rxc ∈ I, we have rx ∈ S. □

Theorem 2.6 Let I be an ideal of R and consider S = \sqrt{I} \ I. If S is a nonempty set, then ⟨S⟩ is connected.

Proof. Let x, y ∈ S. If xy ∈ I, then the result is obtained. Suppose that xy ∉ I, where x^{n}, y^{m} ∈ I and x^{n−1}, y^{m−1} ∉ I. Hence, the path

x — x^{n−1} — xy — y^{m−1} — y

is a path of length four from x to y. □

Corollary 2.7 Suppose either N is the nil radical of R, or is a nilpotent ideal of R. If N is nontrivial, then ⟨N \ {0}⟩ is a connected subgraph of Γ(R).

3 Complete r-Partite Graph

It is easy to see that if I is a prime ideal of R, then we have Γ_{I}(R) = ∅. In the following, we show that if I = 𝔭_{1} ∩ 𝔭_{2}, where 𝔭_{1} and 𝔭_{2} are prime ideals of R, then Γ_{I}(R) is a complete bipartite graph. In section 4, we study the girth and the clique number of Γ_{I}(R) for I = 𝔭_{1} ∩ … ∩ 𝔭_{n}, where 𝔭_{i}’s are prime ideals of R.

Theorem 3.1 Let I be a nonzero ideal of R. Then the following hold:

(a) If 𝔭_{1} and 𝔭_{2} are prime ideals of R and I = 𝔭_{1} ∩ 𝔭_{2} ≠ 0, then Γ_{I}(R) is a complete bipartite graph.

(b) If I ≠ 0 is an ideal of R for which I = \sqrt{I}, then Γ_{I}(R) is a complete bipartite graph if and only if there exist prime ideals 𝔭_{1} and 𝔭_{2} of R such that I = 𝔭_{1} ∩ 𝔭_{2}.

Proof. (a): Let a, b ∈ R \ I with ab ∈ I. Then ab ∈ 𝔭_{1} and ab ∈ 𝔭_{2}. Since 𝔭_{1} and 𝔭_{2} are prime, we have a ∈ 𝔭_{1} or b ∈ 𝔭_{1} and a ∈ 𝔭_{2} or b ∈ 𝔭_{2}. Therefore, suppose a ∈ 𝔭_{1} \ 𝔭_{2} and b ∈ 𝔭_{2} \ 𝔭_{1}. Thus, Γ_{I}(R) is a complete bipartite graph with parts 𝔭_{1} \ 𝔭_{2} and 𝔭_{2} \ 𝔭_{1}.

(b): Suppose that the parts of Γ_{I}(R) are V_{1} and V_{2}. Set 𝔭_{1} = V_{1} ∪ I and 𝔭_{2} = V_{2} ∪ I. It is clear that I = 𝔭_{1} ∩ 𝔭_{2}. We now prove that 𝔭_{1} is an ideal of R. To show this let a, b ∈ 𝔭_{1}.

Case 1: If a, b ∈ I, then a − b ∈ I and so a − b ∈ 𝔭_{1}.

Case 2: If a, b ∈ V_{1}, then there is c ∈ V_{2} such that ca ∈ I and cb ∈ I. So, c(a − b) ∈ I. If a − b ∈ I, then a − b ∈ 𝔭_{1}. Otherwise, a − b ∈ V_{1}, which implies a − b ∈ 𝔭_{1}.

Case 3: If a ∈ V_{1} and b ∈ I, then a − b ∉ I, so there is c ∈ V_{2} such that c(a − b) ∈ I. This implies that a − b ∈ V_{1}, and so a − b ∈ 𝔭_{1}.

Now let r ∈ R and a ∈ 𝔭_{1}.

Case 1: If a ∈ I, then ra ∈ I and so ra ∈ 𝔭_{1}.

Case 2: If a ∈ V_{1}, then there exists c ∈ V_{2} such that ca ∈ I. So, c(ra) ∈ I. If ra ∈ I, then ra ∈ 𝔭_{1} and if ra ∉ I, then ra ∈ V_{1} which implies ra ∈ 𝔭_{1}. Therefore, 𝔭_{1} ⊴ R.

We now prove 𝔭_{1} is prime. For proving this let ab ∈ 𝔭_{1} and a, b ∉ 𝔭_{1}. Since 𝔭_{1} = V_{1} ∪ I, ab ∈ I or ab ∈ V_{1}, and so in any case there exists c ∈ V_{2} such that c(ab) ∈ I. Thus a(cb) ∈ I. If cb ∈ I, then by the definition of Γ_{I}(R) we have b ∈ V_{1}, that is a contradiction. Hence, cb ∉ I and so cb ∈ V_{1}. Therefore, c^{2}b ∈ I. Since I = \sqrt{I}, c^{2} ∉ I. Hence, c^{2} ∈ V_{2}. So b ∈ V_{1} which is a contradiction. Therefore, 𝔭_{1} is a prime ideal of R. □

Note that if we consider R = ℤ_{8} and I = ⟨4⟩, then it is easy to see that Γ_{I}(R) is bipartite, but I cannot be written as the intersection of two prime ideals. Therefore, the converse of Theorem 3.1 (a) is not valid in general. Hence, the condition “I = \sqrt{I}” on ideal I is not superficial in Theorem 3.1 (b).

Theorem 3.2 Let I be a nonzero proper ideal of R. If Γ_{I}(R) is a complete r-partite graph, r ≥ 3, then at most one of the parts has more than one vertex.

Proof. Assume that V_{1}, …, V_{r} are parts of Γ_{I}(R). Let V_{t} and V_{s} have more than one element. Choose x ∈ V_{t} and y ∈ V_{s}. Let V_{l} be a part of Γ_{I}(R) such that V_{l} ≠ V_{t} and V_{l} ≠ V_{s}. Let z ∈ V_{l}. Since Γ_{I}(R) is a complete r-partite graph, (I : x) = (⋃_{1≤i≤r, i≠t} V_{i}) ∪ I, (I : y) = (⋃_{1≤i≤r, i≠s} V_{i}) ∪ I, and (I : z) = (⋃_{1≤i≤r, i≠l} V_{i}) ∪ I. Therefore, (I : z) ⊆ (I : x) ∪ (I : y), and so we have (I : z) ⊆ (I : x) or (I : z) ⊆ (I : y). Let (I : z) ⊆ (I : x) and choose x′ ∈ V_{t} such that x′ ≠ x. Then we have x′ ∈ (I : z) \ (I : x). This is a contradiction. □

4 Girth and Clique Number

In this section we study the girth and the clique number of Γ_{I}(R), when I is an intersection of prime ideals.

Theorem 4.1 Let 𝔭_{1} and 𝔭_{2} be prime ideals of R and I = 𝔭_{1} ∩ 𝔭_{2}. Then either gr(Γ_{I}(R)) = 4 or R/I ≅ ℤ_{2} ⊕ ℤ_{2}.

Proof. If |𝔭_{1} \ 𝔭_{2}| = 1 and |𝔭_{2} \ 𝔭_{1}| ≥ 2, then Γ_{I}(R) is a star graph and so has a cut point. This is a contradiction by [11, Theorem 3.2]. Therefore, this case cannot happen. The case |𝔭_{2} \ 𝔭_{1}| = 1 and |𝔭_{1} \ 𝔭_{2}| ≥ 2 is similar. So there are two other possibilities.

Case 1: |𝔭_{i} \ 𝔭_{j}| ≥ 2 for i ≠ j and 1 ≤ i, j ≤ 2. In this case, Theorem 3.1 implies that gr(Γ_{I}(R)) = 4.

Case 2: |𝔭_{i} \ 𝔭_{j}| < 2 for i ≠ j and 1 ≤ i, j ≤ 2. In this case, there is x ∈ R for which 𝔭_{1} \ 𝔭_{2} = {x} and so 𝔭_{1} = {x} ∪ I. For any r ∈ R \ 𝔭_{2} we have rx ∈ 𝔭_{1} \ I and so rx = x. Therefore, (1 − r)x = 0 ∈ 𝔭_{2} and hence (1 − r) ∈ 𝔭_{2}. Thus |R/𝔭_{2}| = 2. That implies 𝔭_{2} is a maximal ideal of R and R/𝔭_{2} ≅ ℤ_{2}. But 𝔭_{1} + 𝔭_{2} = R, so that implies R/I ≅ R/𝔭_{1} × R/𝔭_{2} ≅ ℤ_{2} ⊕ ℤ_{2} . □

Theorem 4.2 Let I be an ideal of R such that I = ⋂_{1≤i≤n} 𝔭_{i} and for each 1 ≤ j ≤ n, I ≠ ⋂_{1≤i≤n, i≠j} 𝔭_{i} where 𝔭_{i}’s are prime ideals of R. Then ω(Γ_{I}(R)) = n.

Proof. Consider x_{j} ∈ ⋂_{1≤i≤n, i≠j} 𝔭_{i} \ 𝔭_{j}. It is easy to see that X = {x_{1}, …, x_{n}} is a clique in Γ_{I}(R). Hence, ω(Γ_{I}(R)) ≥ n and so it is sufficient to show that ω(Γ_{I}(R)) ≤ n. In order to do this, we use induction on n. For n = 2, by Theorem 3.1, Γ_{I}(R) is a bipartite graph and hence ω(Γ_{I}(R)) = 2. Suppose n > 2 and the result is true for any integer less than n. Let I = ⋂_{1≤i≤n} 𝔭_{i} and for each 1 ≤ j ≤ n, I ≠ ⋂_{1≤i≤n, i≠j} 𝔭_{i}. Let {x_{1}, …, x_{m}} be a clique in Γ_{I}(R). Hence, x_{1}x_{j} ∈ ⋂_{1≤i≤n} 𝔭_{i} for any 2 ≤ j ≤ m. Without loss of generality, suppose that x_{1} ∉ 𝔭_{1}. Therefore, x_{2}, …, x_{m} ∈ 𝔭_{1}, so x_{2}, …, x_{m} ∉ ⋂_{2≤i≤n} 𝔭_{i}. Let J = ⋂_{2≤i≤n} 𝔭_{i}. Hence, {x_{2} …, x_{m}} is a clique in Γ_{J}(R). Therefore, m − 1 ≤ n − 1, and the result is obtained. □

Corollary 4.3 The following hold:

(a) If I = ⋂_{1≤i≤n} 𝔭_{i} ≠ 0 and J = ⋂_{1≤j≤m} 𝔮_{j} where 𝔭_{i}’s and 𝔮_{j}’s are prime ideals such that Γ_{I}(R) = Γ_{J}(R), then m = n.

(b) If for any 𝔭 ∈ Min(R), 𝔭 is a finitely generated ideal, then ω(Γ_{nil(R)}(R)) = |Min(R)| (which is finite by the main theorem of [3]).

(c) If R is a semi-local ring and not local, then ω(Γ_{J(R)}(R)) = |Max(R)|.

(d) If n is a square-free integer, then ω(Γ_{nℤ}(ℤ)) = k, where k is the number of primes in the decomposition of n into primes.

Theorem 4.4 Let I be an ideal of R. Suppose either I is a primary ideal of R that is not prime and |I| ≥ 3, or |Ass(R/I)| ≥ 3. Then gr(Γ_{I}(R)) = 3.

Proof. For the first case, let a, b ∈ R \ I such that ab ∈ I. Then there exists n ∈ ℕ such that b^{n} ∈ I, so we can choose t ∈ ℕ for which b^{t} ∈ I and b^{t−1} ∉ I. Since a, b^{t−1} ∉ I, we have the chain

a — b — b^{t−1} — a

in the graph Γ_{I}(R). Therefore, gr(Γ_{I}(R)) = 3.

For the second case, |Ass(R/I)| ≥ 3 implies that gr(Γ(R/I)) = 3 (see [1, Corollary 2.2]), and hence gr(Γ_{I}(R)) = 3. □

In the above theorem, one of the conditions “|I| ≥ 3” or “|Ass(R/I)| ≥ 3” are necessary. To see this, for example let R = ℤ_{8} and consider I = ⟨4⟩; and note that we have |Ass(R/I)| = 1 and gr(Γ(R/I)) = ∞.

Acknowledgment: This work was done while the second author was a Postdoctoral Research Associate at the School of Mathematics, Institute for Studies in Theoretical Physics and Mathematics (IPM). He would like to thank the IPM for the financial support. Also the authors would like to thank Jim Kuzmanovich and the referee for his/her interest in the subject and making useful suggestions and comments which led to improvement and simplification of the first draft.

References

[1] S. Akbari, H. R. Maimani, S. Yassemi, When a Zero-Divisor Graph is Planar or a Complete r-Partite Graph, J. Algebra 270 (2003), no. 1, 169-180.

[2] S. Akbari, A. Mohammadian, On the Zero-Divisor Graph of a Commutative Ring, J. Algebra 274 (2004), no. 2, 847-855.

[3] D. D. Anderson, A Note on Minimal Prime Ideals, Proc. Amer. Math. Soc. 122 (1994), no. 1, 13-14.

[4] D. F. Anderson, A. Frazier, A. Lauve, P. S. Livingston, “The Zero-Divisor Graph of a Commutative Ring II”, Lecture Notes in Pure and Appl. Math., 220, Dekker, New York, 2001.

[5] D. F. Anderson, R. Levy, J. Shapiro, Zero-Divisor Graphs, von Neumann Regular Rings, and Boolean Algebras, J. Pure Appl. Algebra 180 (2003), no. 3, 221-241.

[6] D. F. Anderson, P. S. Livingston, The Zero-Divisor Graph of a Commutative Ring, J. Algebra 217 (1999), no. 2, 434-447.

[7] I. Beck, Coloring of Commutative Rings, J. Algebra 116 (1988), no. 1, 208-226.

[8] G. Chartrand, O. R. Oellermann, “Applied and Algorithmic Graph Theory”, McGraw-Hill, Inc., New York, 1993.

[9] I. Kaplansky, “Commutative Rings”, The University of Chicago Press, Chicago, Ill.-London, 1974.

[10] R. Levy, J. Shapiro, The Zero-Divisor Graph of von Neumann Regular Rings, Comm. Algebra 30 (2002), no. 2, 745-750.

[11] S. P. Redmond, An Ideal-Based Zero-Divisor Graph of a Commutative Ring, Comm. Algebra 31 (2003), no. 9, 4425-4443.

The Authors’ Addresses

H. R. Maimani, Department of Mathematics, Shahid Rajaee University, P.O. Box 16785-163, Tehran, Iran, and School of Mathematics, Institute for Studies in Theoretical Physics and Mathematics, P.O. Box 19395-5746, Tehran, Iran.

E-mail address: maimani@ipm.ir

M. R. Pournaki, School of Mathematics, Institute for Studies in Theoretical Physics and Mathematics, P.O. Box 19395-5746, Tehran, Iran.

E-mail address: pournaki@ipm.ir

S. Yassemi, Department of Mathematics and Computer Science, Faculty of Science, University of Tehran, Tehran, Iran, and School of Mathematics, Institute for Studies in Theoretical Physics and Mathematics, P.O. Box 19395-5746, Tehran, Iran.

E-mail address: yassemi@ipm.ir