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ISRN Discrete Mathematics
VolumeΒ 2011Β (2011), Article IDΒ 459547, 7 pages
http://dx.doi.org/10.5402/2011/459547
Research Article

Zero-Divisor Graphs with respect to Ideals in Noncommutative Rings

1Department of Mathematics, University of Guilan, P.O. Box 1914, Rasht, Iran
2Department 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.Β Klostermeyer

Copyright Β© 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 𝑅 be a commutative ring and 𝐼 an ideal of 𝑅 . The zero-divisor graph of 𝑅 with respect to 𝐼 , denoted Ξ“ 𝐼 ( 𝑅 ), is the undirected graph whose vertex set is { π‘₯ ∈ 𝑅 ⧡ 𝐼 | π‘₯ 𝑦 ∈ 𝐼 for some 𝑦 ∈ 𝑅 ⧡ 𝐼 } with two distinct vertices π‘₯ and 𝑦 joined by an edge when π‘₯ 𝑦 ∈ 𝐼 . In this paper, we extend the definition of the ideal-based zero-divisor graph to noncommutative rings.

1. Introduction

Throughout 𝑅 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 𝑅 is defined to be the undirected graph whose vertices are the nonzero zero-divisors of 𝑅 , and where π‘₯ βˆ’ 𝑦 is an edge whenever π‘₯ 𝑦 = 0 . This definition is the basis for several further articles [24]) 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 π‘₯ in a noncommutative ring 𝑅 is a zero-divisor if either π‘₯ 𝑦 = 0 or 𝑦 π‘₯ = 0 for some nonzero 𝑦 ∈ 𝑅 . For a ring 𝑅 (not necessarily with multiplicative identity), define a directed graph Ξ“ ( 𝑅 ) whose vertices are the nonzero zero-divisors of 𝑅 , and where π‘₯ β†’ 𝑦 is a directed edge between vertices π‘₯ and 𝑦 if and only if π‘₯ 𝑦 = 0 . If one views each undirected edge π‘₯ βˆ’ 𝑦 as the pair of directed edges π‘₯ β†’ 𝑦 and 𝑦 β†’ π‘₯ , 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 𝑅 , define a graph Ξ“ ( 𝑅 ) whose vertices are the nonzero zero-divisors of 𝑅 and where π‘₯ βˆ’ 𝑦 is an edge if either π‘₯ 𝑦 = 0 or 𝑦 π‘₯ = 0 . One can think of Ξ“ ( 𝑅 ) as the graph Ξ“ ( 𝑅 ) with all directed edges replaced by undirected edges. Given any ring 𝑅 , Ξ“ ( 𝑅 ) 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 𝑅 be a commutative ring. An element π‘Ž ∈ 𝑅 is called prime to an ideal 𝐼 of 𝑅 if π‘Ÿ π‘Ž ∈ 𝐼 (where π‘Ÿ ∈ 𝑅 ) implies that π‘Ÿ ∈ 𝐼 . Denote by 𝑆 ( 𝐼 ) the set of elements of 𝑅 that are not prime to 𝐼 . A proper ideal 𝐼 of 𝑅 is said to be primal if 𝑆 ( 𝐼 ) forms an ideal; this ideal is always a prime ideal, called the adjoint ideal 𝑃 of 𝐼 . In this case, we also say that 𝐼 is a 𝑃 -primal ideal of 𝑅 .

Later in 1956, Barnes in [9], generalized the concept of primal ideals in noncommutative rings. Let 𝐼 be an ideal of 𝑅 and let π‘₯ be an element of 𝑅 . Set 𝐼 π‘₯ βˆ’ 1 = { 𝑦 ∈ 𝑅 ∢ 𝑦 𝑅 π‘₯ βŠ† 𝐼 } . Evidently 𝐼 π‘₯ βˆ’ 1 is an ideal of 𝑅 containing 𝐼 . The element π‘₯ ∈ 𝑅 is not right prime (nrp) to 𝐼 if 𝐼 π‘₯ βˆ’ 1 β‰  𝐼 . Otherwise, π‘₯ is right prime (rp) to 𝐼 . Denote by 𝑆 π‘Ÿ ( 𝐼 ) the set of elements of 𝑅 that are nrp to 𝐼 . The ideal 𝐼 of 𝑅 is called a right primal ideal of 𝑅 if 𝑆 π‘Ÿ ( 𝐼 ) forms an ideal of 𝑅 , which is then termed the adjoint ideal of 𝐼 . The set 𝑆 𝑙 ( 𝐼 ) 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, 𝑃 is a prime ideal of 𝑅 if π‘₯ 𝑅 𝑦 βŠ† 𝑃 implies that π‘₯ or 𝑦 is in 𝑃 . McCoy has shown that this is equivalent to the property that if 𝑃 divides the product of two ideals then 𝑃 must divide at least one of them. An ideal 𝐼 of 𝑅 is said to be a semiprime ideal if, for π‘Ž ∈ 𝑅 , π‘Ž 𝑅 π‘Ž βŠ† 𝐼 implies that π‘Ž ∈ 𝐼 . It is clear that every prime ideal is semiprime. A nonempty subset 𝑀 of 𝑅 is called an π‘š -system if for each pair of elements π‘Ž , 𝑏 ∈ 𝑀 , there is an element π‘₯ ∈ 𝑅 such that π‘Ž π‘₯ 𝑏 ∈ 𝑀 . So a proper ideal 𝑃 of 𝑅 is prime if and only if 𝑅 ⧡ 𝑃 is an π‘š -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 𝐼 be an ideal of 𝑅 . We define a graph Ξ“ 𝑙 𝐼 ( 𝑅 ) with vertices 𝑇 𝑙 ( 𝐼 ) = { π‘₯ ∈ 𝑅 ⧡ 𝐼 ∢ π‘₯ 𝑅 𝑦 βŠ† 𝐼 f o r s o m e 𝑦 ∈ 𝑅 ⧡ 𝐼 } , and where π‘₯ β†’ 𝑦 is a directed edge between distinct vertices π‘₯ and 𝑦 if and only if π‘₯ 𝑅 𝑦 βŠ† 𝐼 . The set 𝑇 π‘Ÿ ( 𝐼 ) and the graph Ξ“ π‘Ÿ 𝐼 ( 𝑅 ) are defined in a similar way.

Definition 2.2. Let 𝐼 be an ideal of 𝑅 . We define a directed graph Ξ“ 𝐼 ( 𝑅 ) with vertices 𝑇 ( 𝐼 ) = { π‘₯ ∈ 𝑅 ⧡ 𝐼 ∢ π‘₯ 𝑅 𝑦 βŠ† 𝐼 o r 𝑦 𝑅 π‘₯ βŠ† 𝐼 f o r s o m e 𝑦 ∈ 𝑅 ⧡ 𝐼 } , and where π‘₯ β†’ 𝑦 is a directed edge between distinct vertices π‘₯ and 𝑦 if and only if π‘₯ 𝑅 𝑦 βŠ† 𝐼 .

Remark 2.3. (1) Suppose we have two graphs 𝐺 1 and 𝐺 2 and suppose that 𝐺 1 has vertex set 𝑉 1 and edge set 𝐸 1 ; and that 𝐺 2 has vertex set 𝑉 2 and edge set 𝐸 2 . The union of the two graphs, written 𝐺 1 βˆͺ 𝐺 2 , will have vertex set 𝑉 1 βˆͺ 𝑉 2 and edge set 𝐸 1 βˆͺ 𝐸 2 . Now assume that 𝐼 is an ideal of the ring 𝑅 . It is easy to see that 𝑇 ( 𝐼 ) = 𝑇 𝑙 ( 𝐼 ) βˆͺ 𝑇 π‘Ÿ ( 𝐼 ) and 𝐸 ( Ξ“ 𝐼 ( 𝑅 ) ) = 𝐸 ( Ξ“ 𝑙 𝐼 ( 𝑅 ) ) βˆͺ 𝐸 ( Ξ“ π‘Ÿ 𝐼 ( 𝑅 ) ) . Therefore, Ξ“ 𝐼 ( 𝑅 ) = Ξ“ 𝑙 𝐼 ( 𝑅 ) βˆͺ Ξ“ π‘Ÿ 𝐼 ( 𝑅 )
(2) We note that if we consider 𝐼 = 0 , and 𝑅 has a two-sided identity, then Ξ“ 0 ( 𝑅 ) = Ξ“ ( 𝑅 ) .

We say that a directed graph 𝐺 is strongly connected if there is a path following the directed edges of 𝐺 from any vertex of 𝐺 to any other vertex of 𝐺 . For two distinct vertices π‘Ž and 𝑏 in a graph 𝐺 , the distance between π‘Ž and 𝑏 , denoted 𝑑 ( π‘Ž , 𝑏 ) , is the length of the shortest path from π‘Ž to 𝑏 if such a path exists; otherwise, 𝑑 ( π‘Ž , 𝑏 ) = ∞ . The diameter of a strongly connected graph is the supremum of the distances between vertices. Redmond proved that if 𝐼 is an ideal of a commutative ring 𝑅 , then the graph Ξ“ 𝐼 ( 𝑅 ) is always connected and its diameter, d i a m ( Ξ“ 𝐼 ( 𝑅 ) ) , is always less than or equal to 3 [5, Theorem 2.4]. The graph Ξ“ 𝐼 ( 𝑅 ) is not in general strongly connected. For example, if we consider the case where ξ€· 𝑅 = { π‘₯ 𝑦 0 0 ξ€Έ ∣ π‘₯ , 𝑦 ∈ β„€ 2 } , and 𝐼 = 0 , then Ξ“ 𝐼 ( 𝑅 ) is not strongly connected as a directed graph (see Figure 1). But we have the following theorem.

459547.fig.001
Figure 1: Ξ“ 𝐼 ( 𝑅 ) , where 𝑅 = { ( π‘₯ 0 𝑦 0 ) ∣ π‘₯ , 𝑦 ∈ β„€ 2 } and 𝐼 = 0 .

Theorem 2.4. Let 𝐼 be an ideal of 𝑅 . If 𝑇 𝑙 ( 𝐼 ) = 𝑇 π‘Ÿ ( 𝐼 ) , then Ξ“ 𝐼 ( 𝑅 ) is strongly connected with d i a m ( Ξ“ 𝐼 ( 𝑅 ) ) ≀ 3 .

Proof. Let π‘₯ and 𝑦 be two distinct vertices of Ξ“ 𝐼 ( 𝑅 ) . Consider the following cases.(1)If π‘₯ 𝑅 𝑦 βŠ† 𝐼 , then π‘₯ β†’ 𝑦 is a path in Ξ“ 𝐼 ( 𝑅 ) .(2)If ΜΈ π‘₯ 𝑅 𝑦 βŠ† 𝐼 , π‘₯ 𝑅 π‘₯ βŠ† 𝐼 and 𝑦 𝑅 𝑦 βŠ† 𝐼 , then, there exists π‘Ÿ ∈ 𝑅 with π‘₯ π‘Ÿ 𝑦 ∈ 𝑅 ⧡ 𝐼 . In this case, π‘₯ β†’ π‘₯ π‘Ÿ 𝑦 β†’ 𝑦 is a path.(3)If ΜΈ π‘₯ 𝑅 𝑦 βŠ† 𝐼 , π‘₯ 𝑅 π‘₯ βŠ† 𝐼 and ΜΈ 𝑦 𝑅 𝑦 βŠ† 𝐼 , then, there exists 𝑏 ∈ 𝑅 ⧡ ( 𝐼 βˆͺ { π‘₯ , 𝑦 } ) with 𝑏 𝑅 𝑦 βŠ† 𝐼 . If π‘₯ 𝑅 𝑏 βŠ† 𝐼 , then π‘₯ β†’ 𝑏 β†’ 𝑦 is a path. If ΜΈ π‘₯ 𝑅 𝑏 βŠ† 𝐼 , there exists π‘Ÿ ∈ 𝑅 with π‘₯ π‘Ÿ 𝑏 βˆ‰ 𝐼 . In this case, π‘₯ β†’ π‘₯ π‘Ÿ 𝑏 β†’ 𝑦 is a path.(4)If ΜΈ π‘₯ 𝑅 𝑦 βŠ† 𝐼 , ΜΈ π‘₯ 𝑅 π‘₯ βŠ† 𝐼 and 𝑦 𝑅 𝑦 βŠ† 𝐼 , then, there exists π‘Ž ∈ 𝑅 ⧡ ( 𝐼 βˆͺ { π‘₯ , 𝑦 } ) such that π‘₯ 𝑅 π‘Ž βŠ† 𝐼 . If π‘Ž 𝑅 𝑦 βŠ† 𝐼 , then π‘₯ β†’ π‘Ž β†’ 𝑦 is a path. If ΜΈ π‘Ž 𝑅 𝑦 βŠ† 𝐼 , there exists π‘Ÿ ∈ 𝑅 with π‘Ž π‘Ÿ 𝑦 βˆ‰ 𝐼 . In this case, π‘₯ β†’ π‘Ž π‘Ÿ 𝑦 β†’ 𝑦 ia a path.(5)If ΜΈ π‘₯ 𝑅 𝑦 βŠ† 𝐼 , ΜΈ π‘₯ 𝑅 π‘₯ βŠ† 𝐼 , and ΜΈ 𝑦 𝑅 𝑦 βŠ† 𝐼 , then, there exist π‘Ž , 𝑏 ∈ 𝑅 ⧡ ( 𝐼 βˆͺ { π‘₯ , 𝑦 } ) with π‘₯ 𝑅 π‘Ž βŠ† 𝐼 and 𝑏 𝑅 𝑦 βŠ† 𝐼 . If π‘Ž = 𝑏 , then π‘₯ β†’ π‘Ž β†’ 𝑦 is a path. If π‘Ž β‰  𝑏 and π‘Ž 𝑅 𝑏 βŠ† 𝐼 , then π‘₯ β†’ π‘Ž β†’ 𝑏 β†’ 𝑦 is a path. If π‘Ž β‰  𝑏 and ΜΈ π‘Ž 𝑅 𝑏 βŠ† 𝐼 , there is π‘Ÿ ∈ 𝑅 with π‘Ž π‘Ÿ 𝑏 βˆ‰ 𝐼 ; in this case, π‘₯ β†’ π‘Ž π‘Ÿ 𝑏 β†’ 𝑦 is a path.

We now define an undirected graph as follows.

Definition 2.5. Let 𝐼 be an ideal of 𝑅 . We define an undirected graph Ξ“ 𝐼 ( 𝑅 ) with vertices 𝑇 ( 𝐼 ) = { π‘₯ ∈ 𝑅 ⧡ 𝐼 ∢ π‘₯ 𝑅 𝑦 βŠ† 𝐼 o r 𝑦 𝑅 π‘₯ βŠ† 𝐼 f o r s o m e 𝑦 ∈ 𝑅 ⧡ 𝐼 } , where distinct vertices π‘₯ and 𝑦 are adjacent if and only if either π‘₯ 𝑅 𝑦 βŠ† 𝐼 or 𝑦 𝑅 π‘₯ βŠ† 𝐼 .

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

Theorem 2.7. Let 𝐼 be an ideal of 𝑅 . Then Ξ“ 𝐼 ( 𝑅 ) is connected with d i a m ( Ξ“ 𝐼 ( 𝑅 ) ) ≀ 3 .

Proof. Let π‘₯ , 𝑦 ∈ Ξ“ 𝐼 ( 𝑅 ) with π‘₯ β‰  𝑦 . If π‘₯ 𝑅 𝑦 βŠ† 𝐼 or 𝑦 𝑅 π‘₯ βŠ† 𝐼 , then π‘₯ βˆ’ 𝑦 is a path and 𝑑 ( π‘₯ , 𝑦 ) = 1 . So assume that ΜΈ π‘₯ 𝑅 𝑦 βŠ† 𝐼 and ΜΈ 𝑦 𝑅 π‘₯ βŠ† 𝐼 . Consider the following cases.
Case 1. π‘₯ 𝑅 π‘₯ βŠ† 𝐼 and 𝑦 𝑅 𝑦 βŠ† 𝐼 . As ΜΈ π‘₯ 𝑅 𝑦 βŠ† 𝐼 , there exists π‘Ÿ ∈ 𝑅 with π‘₯ π‘Ÿ 𝑦 ∈ 𝑅 ⧡ 𝐼 . In this case π‘₯ βˆ’ π‘₯ π‘Ÿ 𝑦 βˆ’ 𝑦 is a path and 𝑑 ( π‘₯ , 𝑦 ) ≀ 2 .
Case 2. π‘₯ 𝑅 π‘₯ βŠ† 𝐼 and ΜΈ 𝑦 𝑅 𝑦 βŠ† 𝐼 . There exists 𝑏 ∈ 𝑅 ⧡ ( 𝐼 βˆͺ { π‘₯ , 𝑦 } ) such that either 𝑏 𝑅 𝑦 βŠ† 𝐼 or 𝑦 𝑅 𝑏 βŠ† 𝐼 . If either 𝑏 𝑅 π‘₯ βŠ† 𝐼 or π‘₯ 𝑅 𝑏 βŠ† 𝐼 , then π‘₯ βˆ’ 𝑏 βˆ’ 𝑦 is a path and 𝑑 ( π‘₯ , 𝑦 ) ≀ 2 . Suppose that ΜΈ 𝑏 𝑅 π‘₯ βŠ† 𝐼 and ΜΈ π‘₯ 𝑅 𝑏 βŠ† 𝐼 . If 𝑦 𝑅 𝑏 βŠ† 𝐼 , then π‘₯ βˆ’ 𝑏 π‘Ÿ 1 π‘₯ βˆ’ 𝑦 is a path for some π‘Ÿ 1 ∈ 𝑅 for which 𝑏 π‘Ÿ 1 π‘₯ ∈ 𝑅 ⧡ 𝐼 . If 𝑦 𝑅 𝑏 βŠ† 𝐼 , then π‘₯ βˆ’ π‘₯ π‘Ÿ 2 𝑏 βˆ’ 𝑦 is a path for some π‘Ÿ 2 ∈ 𝑅 for which π‘₯ π‘Ÿ 2 𝑏 ∈ 𝑅 ⧡ 𝐼 . So in this case 𝑑 ( π‘₯ , 𝑦 ) ≀ 2 .
Case 3. If ΜΈ π‘₯ 𝑅 π‘₯ βŠ† 𝐼 and 𝑦 𝑅 𝑦 βŠ† 𝐼 , a similar argument as in Case 2 shows that there exists a path of length at most 2 between π‘₯ and 𝑦 . SO 𝑑 ( π‘₯ , 𝑦 ) ≀ 2 .
Case 4. ΜΈ π‘₯ 𝑅 π‘₯ βŠ† 𝐼 and ΜΈ 𝑦 𝑅 𝑦 βŠ† 𝐼 . Then, there exist π‘Ž , 𝑏 ∈ 𝑅 ⧡ ( 𝐼 βˆͺ { π‘₯ , 𝑦 } ) such that either π‘Ž 𝑅 π‘₯ βŠ† 𝐼 or π‘₯ 𝑅 π‘Ž βŠ† 𝐼 and such that either 𝑏 𝑅 𝑦 βŠ† 𝐼 or 𝑦 𝑅 𝑏 βŠ† 𝐼 . If π‘Ž = 𝑏 , then π‘₯ βˆ’ π‘Ž βˆ’ 𝑦 is a path and 𝑑 ( π‘₯ , 𝑦 ) ≀ 2 . If π‘Ž β‰  𝑏 and either π‘Ž 𝑅 𝑏 βŠ† 𝐼 or 𝑏 𝑅 π‘Ž βŠ† 𝐼 , then π‘₯ βˆ’ π‘Ž βˆ’ 𝑏 βˆ’ 𝑦 is a path and 𝑠 ( π‘₯ , 𝑦 ) ≀ 3 . So assume that π‘Ž β‰  𝑏 , ΜΈ π‘Ž 𝑅 𝑏 βŠ† 𝐼 , and ΜΈ 𝑏 𝑅 π‘Ž βŠ† 𝐼 . Then we have the following subcases.
Subcase 1. If π‘Ž 𝑅 𝑦 βŠ† 𝐼 or 𝑦 𝑅 π‘Ž βŠ† 𝐼 , then π‘₯ βˆ’ π‘Ž βˆ’ 𝑦 is a path and 𝑑 ( π‘₯ , 𝑦 ) ≀ 2 .
Subcase 2. π‘₯ 𝑅 π‘Ž βŠ† 𝐼 and 𝑏 𝑅 𝑦 βŠ† 𝐼 . As ΜΈ π‘Ž 𝑅 𝑏 βŠ† 𝐼 , there exists π‘Ÿ ∈ 𝑅 with π‘Ž π‘Ÿ 𝑏 ∈ 𝑅 ⧡ 𝐼 . In this case, π‘Ž βˆ’ π‘Ž π‘Ÿ 𝑏 βˆ’ 𝑦 is a path and 𝑑 ( π‘₯ , 𝑦 ) ≀ 2 .
Subcase 3. π‘Ž 𝑅 π‘₯ βŠ† 𝐼 and 𝑦 𝑅 𝑏 βŠ† 𝐼 . As ΜΈ 𝑏 𝑅 π‘Ž βŠ† 𝐼 , there exists π‘Ÿ ∈ 𝑅 with 𝑏 π‘Ÿ π‘Ž ∈ 𝑅 ⧡ 𝐼 . Then π‘₯ βˆ’ 𝑏 π‘Ÿ π‘Ž βˆ’ 𝑦 is a path and 𝑑 ( π‘₯ , 𝑦 ) ≀ 2 .
Subcase 4. If π‘₯ 𝑅 π‘Ž βŠ† 𝐼 , 𝑦 𝑅 𝑏 𝐼 and ΜΈ π‘Ž 𝑅 𝑦 βŠ† 𝐼 , there exists π‘Ÿ ∈ 𝑅 such that π‘Ž π‘Ÿ 𝑦 ∈ 𝑅 ⧡ 𝐼 . In this case, π‘₯ βˆ’ π‘Ž π‘Ÿ 𝑦 βˆ’ 𝑏 βˆ’ 𝑦 is a path and 𝑑 ( π‘₯ , 𝑦 ) ≀ 3 .
Subcase 5. If π‘Ž 𝑅 π‘₯ βŠ† 𝐼 , 𝑏 𝑅 𝑦 βŠ† 𝐼 and ΜΈ 𝑦 𝑅 π‘Ž βŠ† 𝐼 , there exists π‘Ÿ ∈ 𝑅 with 𝑦 π‘Ÿ π‘Ž ∈ 𝑅 ⧡ 𝐼 . Then π‘₯ βˆ’ 𝑦 π‘Ÿ π‘Ž βˆ’ 𝑏 βˆ’ 𝑦 is a path and 𝑑 ( π‘₯ , 𝑦 ) ≀ 3 .
We have already shown that in any case, there exists a path between π‘₯ and 𝑦 and 𝑑 ( π‘₯ , 𝑦 ) ≀ 3 . Thus d i a m ( Ξ“ 𝐼 ( 𝑅 ) ) ≀ 3 .

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

The girth of a graph 𝐺 is the length of a shortest cycle (or equivalently the number of vertices of a least sided polygon) contained in the graph. If 𝐺 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 𝐼 of a commutative ring 𝑅 , the girth of Ξ“ 𝐼 ( 𝑅 ) 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 Ξ“ 𝐼 ( 𝑅 ) .

Theorem 2.8. Let 𝐼 be an ideal of 𝑅 . If Ξ“ 𝐼 ( 𝑅 ) contains a cycle, then 𝑔 π‘Ÿ ( Ξ“ 𝐼 ( 𝑅 ) ) ≀ 4 .

Proof. Suppose that Ξ“ 𝐼 ( 𝑅 ) contains a cycle π‘₯ 0 βˆ’ π‘₯ 1 βˆ’ π‘₯ 2 βˆ’ β‹― βˆ’ π‘₯ 𝑛 βˆ’ 1 βˆ’ π‘₯ 𝑛 βˆ’ π‘₯ 0 of shortest length with 𝑛 β‰₯ 4 and look for a contradiction. Consider the following cases.
Case 1. There exists 1 ≀ 𝑗 ≀ 𝑛 βˆ’ 1 such that π‘₯ 𝑗 ∈ 𝐴 ∢ = 𝐼 π‘₯ βˆ’ 1 𝑗 βˆ’ 1 ∩ 𝐼 π‘₯ βˆ’ 1 𝑗 + 1 . Without loss of generality, we may assume that 𝑗 = 1 . If there exists 𝑦 ∈ 𝐴 ⧡ 𝐼 with 𝑦 β‰  π‘₯ 1 , then π‘₯ 0 βˆ’ π‘₯ 1 βˆ’ π‘₯ 2 βˆ’ 𝑦 βˆ’ π‘₯ 0 is a cycle in Ξ“ 𝐼 ( 𝑅 ) which is a contradiction. So assume that 𝐴 = 𝐼 βˆͺ { π‘₯ 1 } . Since π‘₯ 3 βˆ’ π‘₯ 4 is a path, either π‘₯ 3 𝑅 π‘₯ 4 βŠ† 𝐼 or π‘₯ 4 𝑅 π‘₯ 3 βŠ† 𝐼 . Note that π‘₯ 1 𝑅 π‘₯ 3 ΜΈ βŠ† 𝐼 , π‘₯ 3 𝑅 π‘₯ 1 ΜΈ βŠ† 𝐼 , π‘₯ 1 𝑅 π‘₯ 4 ΜΈ βŠ† 𝐼 , and π‘₯ 4 𝑅 π‘₯ 1 ΜΈ βŠ† 𝐼 . As 𝐴 is an ideal of 𝑅 and π‘₯ 1 ∈ 𝐴 , we have π‘₯ 3 𝑅 π‘₯ 1 βŠ† 𝐴 = 𝐼 βˆͺ { π‘₯ 1 } . There exists π‘Ÿ ∈ 𝑅 with π‘₯ 3 π‘Ÿ π‘₯ 1 ∈ 𝑅 ⧡ 𝐼 . But π‘₯ 3 π‘Ÿ π‘₯ 1 ∈ 𝐴 implies that π‘₯ 3 π‘Ÿ π‘₯ 1 = π‘₯ 1 . Similarly, one can shows that π‘₯ 4 π‘Ÿ ξ…ž π‘₯ 1 = π‘₯ 1 for some π‘Ÿ ξ…ž ∈ 𝑅 . In this case, we have π‘₯ 3 𝑅 π‘₯ 1 = π‘₯ 3 𝑅 π‘₯ 4 π‘Ÿ ξ…ž π‘₯ 1 βŠ† 𝐼 which is a contradiction.
Case 2. There exists 1 ≀ 𝑗 ≀ 𝑛 βˆ’ 1 with π‘₯ 𝑗 ∈ π‘₯ βˆ’ 1 𝑗 βˆ’ 1 𝐼 ∩ π‘₯ βˆ’ 1 𝑗 + 1 𝐼 . A similar argument as in Case 1 leads us a contradiction.
Case 3. For each 1 ≀ 𝑗 ≀ 𝑛 βˆ’ 1 , π‘₯ 𝑗 βˆ‰ 𝐼 π‘₯ βˆ’ 1 𝑗 βˆ’ 1 ∩ 𝐼 π‘₯ βˆ’ 1 𝑗 + 1 , and π‘₯ 𝑗 ∈ π‘₯ βˆ’ 1 𝑗 βˆ’ 1 𝐼 ∩ π‘₯ βˆ’ 1 𝑗 + 1 𝐼 . Therefore, without loss of generality, we can assume that we have a cycle in Ξ“ 𝐼 ( 𝑅 ) of the form π‘₯ 0 β†’ π‘₯ 1 β†’ π‘₯ 2 β†’ β‹― β†’ π‘₯ 𝑛 βˆ’ 1 β†’ π‘₯ 𝑛 β†’ π‘₯ 0 with all edges having only one direction. Now consider the following subcases.
Subcase 1. π‘₯ 0 𝑅 π‘₯ 0 βŠ† 𝐼 and π‘₯ 𝑛 𝑅 π‘₯ 𝑛 βŠ† 𝐼 . Since there is no directed path of the form π‘₯ 0 β†’ π‘₯ 𝑛 , there exists π‘Ÿ ∈ 𝑅 with π‘₯ 0 π‘Ÿ π‘₯ 𝑛 ∈ 𝑅 ⧡ 𝐼 . In this case, π‘₯ 0 β†’ π‘₯ 0 π‘Ÿ π‘₯ 𝑛 β†’ π‘₯ 𝑛 β†’ π‘₯ 0 is a 3-cycle in Ξ“ 𝐼 ( 𝑅 ) which is a contradiction.
Subcase 2. π‘₯ 0 𝑅 π‘₯ 0 βŠ† 𝐼 and π‘₯ 𝑛 𝑅 π‘₯ 𝑛 ΜΈ βŠ† 𝐼 . Since 𝑛 β‰₯ 4 , π‘₯ 0 𝑅 π‘₯ 𝑛 βˆ’ 1 ΜΈ βŠ† 𝐼 . So there is π‘Ÿ ∈ 𝑅 with π‘₯ 0 π‘Ÿ π‘₯ 𝑛 βˆ’ 1 ∈ 𝑅 ⧡ 𝐼 . In this case, π‘₯ 0 β†’ π‘₯ 0 π‘Ÿ π‘₯ 𝑛 βˆ’ 1 β†’ π‘₯ 𝑛 β†’ π‘₯ 0 is a 3-cycle in Ξ“ 𝐼 ( 𝑅 ) which is a contradiction.
Subcase 3. π‘₯ 0 𝑅 π‘₯ 0 ΜΈ βŠ† 𝐼 and π‘₯ 𝑛 𝑅 π‘₯ 𝑛 βŠ† 𝐼 . Since 𝑛 β‰₯ 4 , π‘₯ 1 𝑅 π‘₯ 𝑛 ΜΈ βŠ† 𝐼 . So there is π‘Ÿ ∈ 𝑅 with π‘₯ 1 π‘Ÿ π‘₯ 𝑛 ∈ 𝑅 ⧡ 𝐼 . In this case, π‘₯ 0 β†’ π‘₯ 1 π‘Ÿ π‘₯ 𝑛 β†’ π‘₯ 𝑛 β†’ π‘₯ 0 is a 3-cycle in Ξ“ 𝐼 ( 𝑅 ) which is a contradiction.
Subcase 4. π‘₯ 0 𝑅 π‘₯ 0 ΜΈ βŠ† 𝐼 and π‘₯ 𝑛 𝑅 π‘₯ 𝑛 ΜΈ βŠ† 𝐼 . Since 𝑛 β‰₯ 4 , π‘₯ 1 𝑅 π‘₯ 𝑛 βˆ’ 1 ΜΈ βŠ† 𝐼 . So there is π‘Ÿ ∈ 𝑅 with π‘₯ 1 π‘Ÿ π‘₯ 𝑛 βˆ’ 1 ∈ 𝑅 ⧡ 𝐼 . In this case, π‘₯ 0 β†’ π‘₯ 1 π‘Ÿ π‘₯ 𝑛 βˆ’ 1 β†’ π‘₯ 𝑛 β†’ π‘₯ 0 is a 3-cycle in Ξ“ 𝐼 ( 𝑅 ) which is a contradiction.
Since in each of these cases we have found a contradiction, we must have 𝑔 π‘Ÿ ( Ξ“ 𝐼 ( 𝑅 ) ) ≀ 4 .

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 𝐼 be an ideal of 𝑅 , and let π‘₯ be an element of 𝑅 . Set 𝐼 π‘₯ βˆ’ 1 = { 𝑦 ∈ 𝑅 ∢ 𝑦 𝑅 π‘₯ βŠ† 𝐼 } and π‘₯ βˆ’ 1 𝐼 = { 𝑦 ∈ 𝑅 ∢ π‘₯ 𝑅 𝑦 βŠ† 𝐼 } . Evidently both 𝐼 π‘₯ βˆ’ 1 and π‘₯ βˆ’ 1 𝐼 are ideals of 𝑅 containing 𝐼 .

Definition 3.2 (see [9]). The element π‘₯ ∈ 𝑅 is not right prime (nrp) (resp., not left prime (nlp)) to 𝐼 if 𝐼 π‘₯ βˆ’ 1 β‰  𝐼 (resp., π‘₯ βˆ’ 1 𝐼 β‰  𝐼 ). Otherwise, π‘₯ is right prime (rp) (resp., left prime (lp)) to 𝐼 . Denote by 𝑆 π‘Ÿ ( 𝐼 ) (resp., 𝑆 𝑙 ( 𝐼 ) ) the set of elements of 𝑅 that are nrp (resp., nlp) to 𝐼 . The ideal 𝐼 of 𝑅 is called a right primal (resp., left primal) ideal of 𝑅 if 𝑆 π‘Ÿ ( 𝐼 ) (resp., 𝑆 𝑙 ( 𝐼 ) ) form an ideal of 𝑅 , which is then termed the right (resp., left) adjoint ideal of 𝐼 .

Definition 3.3. Let 𝐼 be an ideal of 𝑅 . The element π‘₯ ∈ 𝑅 is not prime (np) to 𝐼 if either 𝐼 π‘₯ βˆ’ 1 β‰  𝐼 or π‘₯ βˆ’ 1 𝐼 β‰  𝐼 . Otherwise, π‘₯ is prime to 𝐼 . Denote by 𝑆 ( 𝐼 ) the set of elements of 𝑅 that are np to 𝐼 . Clearly 𝑆 ( 𝐼 ) = 𝑆 𝑙 ( 𝐼 ) βˆͺ 𝑆 π‘Ÿ ( 𝐼 ) . 𝐼 is called a primal ideal of 𝑅 if 𝑆 ( 𝐼 ) form an ideal of 𝑅 , which is then termed the adjoint ideal of 𝑅 .

Example 3.4 (see [11]). Take 𝑅 = β„€ 2 ⟨ π‘₯ , 𝑦 ⟩ , the noncommutative polynomial ring over β„€ 2 = { 0 , 1 } , subject to π‘₯ 𝑦 = 0 , π‘₯ 𝑅 𝑦 = 0 . It is easy to check that 𝑦 is nrp to the ideal ( 0 ) , and 𝑆 π‘Ÿ ( 0 ) = 𝑅 𝑦 𝑅 . Moreover, for every 𝑧 ∈ 𝑅 , 𝑦 𝑅 𝑧 = 0 if and only if 𝑧 = 0 . Hence 𝑦 is nlp to ( 0 ) . But as π‘₯ 𝑅 𝑦 = 0 with 𝑦 βˆ‰ ( 0 ) , π‘₯ is nlp to ( 0 ) . Therefore, 𝑆 𝑙 ( 0 ) = 𝑅 π‘₯ 𝑅 . These show that ( 0 ) is both left and right primal ideal, but not a primal ideal.

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

Lemma 3.6. Let 𝐼 be a proper ideal of 𝑅 . Then we have the following. (1)All 𝑆 π‘Ÿ ( 𝐼 ) , 𝑆 𝑙 ( 𝐼 ) , and 𝑆 ( 𝐼 ) contain 𝐼 .(2) Ξ“ 𝑙 𝐼 ( 𝑅 ) = 𝑆 𝑙 ( 𝐼 ) ⧡ 𝐼 . In particular, Ξ“ 𝑙 𝐼 ( 𝑅 ) βˆͺ 𝐼 = 𝑆 𝑙 ( 𝐼 ) .(3) Ξ“ π‘Ÿ 𝐼 ( 𝑅 ) = 𝑆 π‘Ÿ ( 𝐼 ) ⧡ 𝐼 . In particular, Ξ“ π‘Ÿ 𝐼 ( 𝑅 ) βˆͺ 𝐼 = 𝑆 π‘Ÿ ( 𝐼 ) .(4) Ξ“ 𝐼 ( 𝑅 ) = 𝑆 ( 𝐼 ) ⧡ 𝐼 . In particular, Ξ“ 𝐼 ( 𝑅 ) βˆͺ 𝐼 = 𝑆 ( 𝐼 ) .

Proof. (1) Assume that 𝑦 ∈ 𝑅 ⧡ 𝐼 . For every π‘₯ ∈ 𝐼 , as π‘₯ 𝑅 𝑦 βŠ† 𝐼 , we must have that π‘₯ is nrp to 𝐼 , that is π‘₯ ∈ 𝑆 π‘Ÿ ( 𝐼 ) . Hence 𝐼 βŠ† 𝑆 π‘Ÿ ( 𝐼 ) . Similarly, one can show that 𝐼 βŠ† 𝑆 𝑙 ( 𝐼 ) and 𝐼 βŠ† 𝑆 ( 𝐼 )
(2) Let π‘₯ ∈ Ξ“ 𝑙 𝐼 ( 𝑅 ) . Then π‘₯ βˆ‰ 𝐼 and π‘₯ 𝑅 𝑦 βŠ† 𝐼 for some 𝑦 ∈ 𝑅 ⧡ 𝐼 . So π‘₯ is nlp to 𝐼 ; hence π‘₯ ∈ 𝑆 𝑙 ( 𝐼 ) ⧡ 𝐼 . Thus Ξ“ 𝑙 𝐼 ( 𝑅 ) βŠ† 𝑆 𝑙 ( 𝐼 ) βˆ’ 𝐼 . For the other containment, assume that π‘Ž ∈ 𝑆 𝑙 ( 𝐼 ) βˆ’ 𝐼 . As π‘Ž is nlp to 𝐼 , there exists 𝑦 βˆ‰ 𝐼 such that π‘Ž 𝑅 𝑦 βŠ† 𝐼 . Thus π‘Ž ∈ Ξ“ 𝑙 𝐼 ( 𝑅 ) , so 𝑆 𝑙 ( 𝐼 ) βˆ’ 𝐼 βŠ† Ξ“ 𝑙 𝐼 ( 𝑅 ) . Therefore, we have the equality.
(3) The proof of ( 3 ) is similar to that of ( 2 ) .
(4) By ( 2 ) and ( 3 ) , we have Ξ“ 𝐼 ( 𝑅 ) = Ξ“ 𝑙 𝐼 ( 𝑅 ) βˆͺ Ξ“ π‘Ÿ 𝐼 ( 𝑅 ) = ( 𝑆 𝑙 ( 𝐼 ) ⧡ 𝐼 ) βˆͺ ( 𝑆 π‘Ÿ ( 𝐼 ) ⧡ 𝐼 ) = ( 𝑆 𝑙 ( 𝐼 ) βˆͺ 𝑆 π‘Ÿ ( 𝐼 ) ) ⧡ 𝐼 = 𝑆 ( 𝐼 ) ⧡ 𝐼 .

Theorem 3.7. Assume that 𝑅 satisfies the ascending chain condition for ideals, and let 𝐼 and 𝑃 be ideals of 𝑅 with 𝑃 prime. Then (1) 𝐼 is a prime ideal of 𝑅 if and only if Ξ“ 𝐼 ( 𝑅 ) = βˆ… ,(2) 𝐼 is a right 𝑃 -primal ideal of 𝑅 if and only if 𝐼 βŠ† 𝑃 and Ξ“ π‘Ÿ 𝐼 ( 𝑅 ) = 𝑃 ⧡ 𝐼 ;(3) 𝐼 is a left 𝑃 -primal ideal of 𝑅 if and only if 𝐼 βŠ† 𝑃 and Ξ“ 𝑙 𝐼 ( 𝑅 ) = 𝑃 ⧡ 𝐼 ;(4) 𝐼 is a 𝑃 -primal ideal of 𝑅 if and only if 𝐼 βŠ† 𝑃 and Ξ“ 𝐼 ( 𝑅 ) = 𝑃 ⧡ 𝐼 ;

Proof. (1) Is clear.
(2) If 𝐼 is right 𝑃 -primal, then 𝑃 = 𝑆 π‘Ÿ ( 𝐼 ) . Hence 𝐼 βŠ† 𝑆 π‘Ÿ ( 𝐼 ) = 𝑃 and Ξ“ π‘Ÿ 𝐼 ( 𝑅 ) = 𝑃 ⧡ 𝐼 by Lemma 3.6. Conversely, if Ξ“ π‘Ÿ 𝐼 ( 𝑅 ) = 𝑃 ⧡ 𝐼 , then 𝑃 = Ξ“ π‘Ÿ 𝐼 ( 𝑅 ) βˆͺ 𝐼 = 𝑆 π‘Ÿ ( 𝐼 ) by Lemma 3.6. Thus 𝐼 is a right 𝑃 -primal ideal of 𝑅 .
The proofs of parts ( 3 ) and ( 4 ) are similar.

Let 𝐼 be an ideal of 𝑅 . The prime radical of 𝐼 , denoted by R a d ( 𝐼 ) , is the set of all π‘Ž ∈ 𝑅 such that the intersection of 𝐼 with every π‘š -system of 𝑅 which contains π‘Ž is nonempty. An ideal 𝑄 of 𝑅 is a primary ideal if 𝐴 𝐡 βŠ† 𝑄 and 𝐡 ΜΈ βŠ† 𝑄 , where 𝐴 and 𝐡 are ideals of 𝑅 , implies that 𝐴 βŠ† R a d ( 𝑄 ) . It was shown that an ideal 𝑄 of 𝑅 is a primary ideal if and only if π‘Ž 𝑅 𝑏 βŠ† 𝑄 and 𝑏 βˆ‰ 𝑄 , where π‘Ž , 𝑏 ∈ 𝑅 implies that π‘Ž ∈ R a d ( 𝑄 ) . The following theorem is another characterization via zero-divisor graphs.

Theorem 3.8. Let 𝑄 be an ideal of 𝑅 . Then 𝑄 is a primary ideal of 𝑅 if and only if Ξ“ 𝑄 ( 𝑅 ) = R a d ( 𝑄 ) ⧡ 𝑄 .

Proof. If 𝑄 is primary, then 𝑄 is a primal ideal of 𝑅 with adjoint ideal R a d ( 𝑄 ) . Thus Ξ“ 𝐼 ( 𝑅 ) = R a d ( 𝑄 ) ⧡ 𝑄 by Lemma 3.6. Conversely, assume that Ξ“ 𝐼 ( 𝑅 ) = R a d ( 𝑄 ) ⧡ 𝑄 and let π‘Ž 𝑅 𝑏 βŠ† 𝑄 for some π‘Ž , 𝑏 ∈ 𝑅 . Assume that π‘Ž , 𝑏 ∈ 𝑅 ⧡ 𝑄 . Then π‘Ž ∈ Ξ“ 𝐼 ( 𝑅 ) = R a d ( 𝑄 ) ⧡ 𝑄 , that is, 𝑄 is a primary ideal.

Acknowledgment

The authors thank the referee for valuable comments.

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