ISRN Discrete MathematicsVolume 2011 (2011), Article ID 459547, 7 pagesdoi:10.5402/2011/459547

Research Article

Zero-Divisor Graphs with respect to Ideals in Noncommutative Rings

Shahabaddin Ebrahimi Atani¹ and Ahmad Yousefian Darani²

¹Department of Mathematics, University of Guilan, P.O. Box 1914, Rasht, Iran²Department of Mathematics, University of Mohaghegh Ardabili, P.O. Box 179, Ardabil 56199-11367, Iran

Received 15 July 2011; Accepted 25 August 2011

Academic Editor: W. F. Klostermeyer

Copyright © 2011 Shahabaddin Ebrahimi Atani and Ahmad Yousefian Darani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Let be a commutative ring and an ideal of . The zero-divisor graph of with respect to , denoted (), is the undirected graph whose vertex set is for some with two distinct vertices and joined by an edge when . In this paper, we extend the definition of the ideal-based zero-divisor graph to noncommutative rings.

1. Introduction

Throughout will denote an associative ring which will be noncommutative unless otherwise specified. The term ideal will always mean two-sided ideal.

In [1], the zero-divisor graph of a commutative ring is defined to be the undirected graph whose vertices are the nonzero zero-divisors of , and where is an edge whenever . This definition is the basis for several further articles [2–4]) examining the relationship between the algebraic structure of a ring and the nature of the resulting graph. The zero-divisor graph has been extended to other algebraic structures in [5, 6].

In [7], this concept was generalized to noncommutative rings in two different ways. An element in a noncommutative ring is a zero-divisor if either or for some nonzero . For a ring (not necessarily with multiplicative identity), define a directed graph whose vertices are the nonzero zero-divisors of , and where is a directed edge between vertices and if and only if . If one views each undirected edge as the pair of directed edges and , then this definition agrees with the above for a commutative ring.

The second definition of the zero-divisor graph introduced in [7] produces an undirected graph. For a ring , define a graph whose vertices are the nonzero zero-divisors of and where is an edge if either or . One can think of as the graph with all directed edges replaced by undirected edges. Given any ring , is connected [7, Theorem 3.2]. (Note that a vertex is never considered adjacent to itself in any of these definitions.)

Fuchs in [8] introduced and studied primal ideals in a commutative ring. Let be a commutative ring. An element is called prime to an ideal of if (where ) implies that . Denote by the set of elements of that are not prime to . A proper ideal of is said to be primal if forms an ideal; this ideal is always a prime ideal, called the adjoint ideal of . In this case, we also say that is a -primal ideal of .

Later in 1956, Barnes in [9], generalized the concept of primal ideals in noncommutative rings. Let be an ideal of and let be an element of . Set . Evidently is an ideal of containing . The element is not right prime (nrp) to if . Otherwise, is right prime (rp) to . Denote by the set of elements of that are nrp to . The ideal of is called a right primal ideal of if forms an ideal of , which is then termed the adjoint ideal of . The set and the concept of left primal ideals is defined in a similar way.

By a prime ideal we mean an ideal which is prime in the sense of McCoy [10], that is, is a prime ideal of if implies that or is in . McCoy has shown that this is equivalent to the property that if divides the product of two ideals then must divide at least one of them. An ideal of is said to be a semiprime ideal if, for , implies that . It is clear that every prime ideal is semiprime. A nonempty subset of is called an -system if for each pair of elements , there is an element such that . So a proper ideal of is prime if and only if is an -system.

In Section 2, we give two definitions of zero-divisors graphs with respect to an ideal in a noncommutative ring, and we study the most basic results on the structure of these graphs. In Section 3, we discuss these graphs with respect to primal ideals.

2. Basic Results

In this section, we define several graphs with respect to an ideal in a noncommutative ring.

Definition 2.1. Let be an ideal of . We define a graph with vertices , and where is a directed edge between distinct vertices and if and only if . The set and the graph are defined in a similar way.

Definition 2.2. Let be an ideal of . We define a directed graph with vertices , and where is a directed edge between distinct vertices and if and only if .

Remark 2.3. (1) Suppose we have two graphs and and suppose that has vertex set and edge set ; and that has vertex set and edge set . The union of the two graphs, written , will have vertex set and edge set . Now assume that is an ideal of the ring . It is easy to see that and . Therefore, (2) We note that if we consider , and has a two-sided identity, then .

We say that a directed graph is strongly connected if there is a path following the directed edges of from any vertex of to any other vertex of . For two distinct vertices and in a graph , the distance between and , denoted , is the length of the shortest path from to if such a path exists; otherwise, . The diameter of a strongly connected graph is the supremum of the distances between vertices. Redmond proved that if is an ideal of a commutative ring , then the graph is always connected and its diameter, , is always less than or equal to 3 [5, Theorem 2.4]. The graph is not in general strongly connected. For example, if we consider the case where , and , then is not strongly connected as a directed graph (see Figure 1). But we have the following theorem.

Figure 1: , where and .

Theorem 2.4. Let be an ideal of . If , then is strongly connected with .

Proof. Let and be two distinct vertices of . Consider the following cases.(1)If , then is a path in .(2)If , and , then, there exists with . In this case, is a path.(3)If , and , then, there exists with . If , then is a path. If , there exists with . In this case, is a path.(4)If , and , then, there exists such that . If , then is a path. If , there exists with . In this case, ia a path.(5)If , , and , then, there exist with and . If , then is a path. If and , then is a path. If and , there is with ; in this case, is a path.

We now define an undirected graph as follows.

Definition 2.5. Let be an ideal of . We define an undirected graph with vertices , where distinct vertices and are adjacent if and only if either or .

Remark 2.6. Note that the graphs and share the same vertices and the same edges if the directions on the edges are ignored. Hence, the only difference between and is that the former one is a directed graph while the latter one is undirected. If is a commutative ring, then this definition agrees with the ideal-based zero-divisor graph in the sense of Redmond.

Theorem 2.7. Let be an ideal of . Then is connected with .

Proof. Let with . If or , then is a path and . So assume that and . Consider the following cases.Case 1. and . As , there exists with . In this case is a path and .Case 2. and . There exists such that either or . If either or , then is a path and . Suppose that and . If , then is a path for some for which . If , then is a path for some for which . So in this case .Case 3. If and , a similar argument as in Case 2 shows that there exists a path of length at most 2 between and . SO .Case 4. and . Then, there exist such that either or and such that either or . If , then is a path and . If and either or , then is a path and . So assume that , , and . Then we have the following subcases.Subcase 1. If or , then is a path and .Subcase 2. and . As , there exists with . In this case, is a path and .Subcase 3. and . As , there exists with . Then is a path and .Subcase 4. If , and , there exists such that . In this case, is a path and .Subcase 5. If , and , there exists with . Then is a path and .We have already shown that in any case, there exists a path between and and . Thus .

As we mentioned in Figure 1, if is a noncommutative ring, the graph need not be strongly connected as a directed graph, while as it is proved in Theorem 2.7, is always connected.

The girth of a graph is the length of a shortest cycle (or equivalently the number of vertices of a least sided polygon) contained in the graph. If does not contain a cycle, then its girth is defined to be infinity. Obviously, the girth of a graph is at least 3. For an ideal of a commutative ring , the girth of is known to be either infinite or 3 or 4 (See [5, Lemma 5.1]). In the following theorem, we give a similar result for .

Theorem 2.8. Let be an ideal of . If contains a cycle, then .

Proof. Suppose that contains a cycle of shortest length with and look for a contradiction. Consider the following cases.Case 1. There exists such that . Without loss of generality, we may assume that . If there exists with , then is a cycle in which is a contradiction. So assume that . Since is a path, either or . Note that , , , and . As is an ideal of and , we have . There exists with . But implies that . Similarly, one can shows that for some . In this case, we have which is a contradiction.Case 2. There exists with . A similar argument as in Case 1 leads us a contradiction.Case 3. For each , , and . Therefore, without loss of generality, we can assume that we have a cycle in of the form with all edges having only one direction. Now consider the following subcases.Subcase 1. and . Since there is no directed path of the form , there exists with . In this case, is a 3-cycle in which is a contradiction.Subcase 2. and . Since , . So there is with . In this case, is a 3-cycle in which is a contradiction.Subcase 3. and . Since , . So there is with . In this case, is a 3-cycle in which is a contradiction.Subcase 4. and . Since , . So there is with . In this case, is a 3-cycle in which is a contradiction. Since in each of these cases we have found a contradiction, we must have .

3. Primal Ideals

In this section, we will study the zero-divisor graphs with respect to primal ideals, right primal ideals, and left primal ideals. First we recall the definitions of these concepts.

Definition 3.1 (see [9]). Let be an ideal of , and let be an element of . Set and . Evidently both and are ideals of containing .

Definition 3.2 (see [9]). The element is not right prime (nrp) (resp., not left prime (nlp)) to if (resp., ). Otherwise, is right prime (rp) (resp., left prime (lp)) to . Denote by (resp., ) the set of elements of that are nrp (resp., nlp) to . The ideal of is called a right primal (resp., left primal) ideal of if (resp., ) form an ideal of , which is then termed the right (resp., left) adjoint ideal of .

Definition 3.3. Let be an ideal of . The element is not prime (np) to if either or . Otherwise, is prime to . Denote by the set of elements of that are np to . Clearly . is called a primal ideal of if form an ideal of , which is then termed the adjoint ideal of .

Example 3.4 (see [11]). Take , the noncommutative polynomial ring over , subject to , . It is easy to check that is nrp to the ideal , and . Moreover, for every , if and only if . Hence is nlp to . But as with , is nlp to . Therefore, . These show that is both left and right primal ideal, but not a primal ideal.

Remark 3.5. (1) Note that if is commutative, then being nrp to is equivalent to being nlp and both are equivalent to being not prime to , and thus the definitions of Barnes [9] and Fuchs [8] are identical.(2) If satisfies the ascending chain condition for ideals and if is a right primal (resp. left primal) ideal of , then, by [9, Corollary 2], the set (resp. ) is a prime ideal of . In this case, we also say that is a right (resp. left) -primal ideal of .

Lemma 3.6. Let be a proper ideal of . Then we have the following. (1)All , , and contain .(2). In particular, .(3). In particular, .(4). In particular, .

Proof. (1) Assume that . For every , as , we must have that is nrp to , that is . Hence . Similarly, one can show that and (2) Let . Then and for some . So is nlp to ; hence . Thus . For the other containment, assume that . As is nlp to , there exists such that . Thus , so . Therefore, we have the equality.(3) The proof of is similar to that of .(4) By and , we have .

Theorem 3.7. Assume that satisfies the ascending chain condition for ideals, and let and be ideals of with prime. Then (1) is a prime ideal of if and only if ,(2) is a right -primal ideal of if and only if and ;(3) is a left -primal ideal of if and only if and ;(4) is a -primal ideal of if and only if and ;

Proof. (1) Is clear.(2) If is right -primal, then . Hence and by Lemma 3.6. Conversely, if , then by Lemma 3.6. Thus is a right -primal ideal of .The proofs of parts and are similar.

Let be an ideal of . The prime radical of , denoted by , is the set of all such that the intersection of with every -system of which contains is nonempty. An ideal of is a primary ideal if and , where and are ideals of , implies that . It was shown that an ideal of is a primary ideal if and only if and , where implies that . The following theorem is another characterization via zero-divisor graphs.

Theorem 3.8. Let be an ideal of . Then is a primary ideal of if and only if .

Proof. If is primary, then is a primal ideal of with adjoint ideal . Thus by Lemma 3.6. Conversely, assume that and let for some . Assume that . Then , that is, is a primary ideal.

Acknowledgment

The authors thank the referee for valuable comments.

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