# Zero-Divisor Graphs with respect to Ideals in Noncommutative Rings

Authors: Shahabaddin Ebrahimi Atani; Ahmad Yousefian Darani
Journal: ISRN Discrete Mathematics
 (2011)

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## Abstract

LetR be a commutative ring and  I an ideal of R. The zero-divisor
graph of R with respect to I, denoted ΓI(R), is the undirected graph whose
vertex set is {x∈R∖I|xy∈I for some y∈R∖I} with two distinct vertices x
and y joined by an edge when xy∈I. In this paper, we extend the definition
of the ideal-based zero-divisor graph to noncommutative rings.


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## Body

## 1. Introduction

ThroughoutR 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 x-y is an edge whenever xy=0. 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 x in a noncommutative ring R is a zero-divisor if either xy=0 or yx=0 for some nonzero y∈R. For a ring R (not necessarily with multiplicative identity), define a directed graph Γ(R) whose vertices are the nonzero zero-divisors of R, and where x→y is a directed edge between vertices x and y if and only if xy=0. If one views each undirected edge x-y as the pair of directed edges x→y and y→x, 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 Γ¯(R) whose vertices are the nonzero zero-divisors of R and where x-y is an edge if either xy=0 or yx=0. One can think of Γ¯(R) as the graph Γ(R) with all directed edges replaced by undirected edges. Given any ring R, Γ¯(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 a∈R is called prime to an ideal I of R if ra∈I (where r∈R) implies that r∈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 x be an element of R. Set Ix-1={y∈R:yRx⊆I}. Evidently Ix-1 is an ideal of R containing I. The element x∈R is not right prime (nrp) to I if Ix-1≠I. Otherwise, x is right prime (rp) to I. Denote by Sr(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 Sr(I) forms an ideal of R, which is then termed the adjoint ideal of I. The set Sl(I) and the concept of left primal ideals is defined in a similar way.By aprime ideal we mean an ideal which is prime in the sense of McCoy [10], that is, P is a prime ideal of R if xRy⊆P implies that x 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 a∈R, aRa⊆I implies that a∈I. It is clear that every prime ideal is semiprime. A nonempty subset M of R is called an m-system if for each pair of elements a,b∈M, there is an element x∈R such that axb∈M. So a proper ideal P of R is prime if and only if R∖P is an m-system.In Section2, 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.
LetI be an ideal of R. We define a graph ΓIl(R) with vertices Tl(I)={x∈R∖I:xRy⊆Ifor somey∈R∖I}, and where x→y is a directed edge between distinct vertices x and y if and only if xRy⊆I. The set Tr(I) and the graph ΓIr(R) are defined in a similar way.Definition 2.2.
LetI be an ideal of R. We define a directed graph ΓI(R) with vertices T(I)={x∈R∖I:xRy⊆IoryRx⊆Ifor somey∈R∖I}, and where x→y is a directed edge between distinct vertices x and y if and only if xRy⊆I.Remark 2.3.
(1) Suppose we have two graphsG1 and G2 and suppose that G1 has vertex set V1 and edge set E1; and that G2 has vertex set V2 and edge set E2. The union of the two graphs, written G1∪G2, will have vertex set V1∪V2 and edge set E1∪E2. Now assume that I is an ideal of the ring R. It is easy to see that T(I)=Tl(I)∪Tr(I) and E(ΓI(R))=E(ΓIl(R))∪E(ΓIr(R)). Therefore, ΓI(R)=ΓIl(R)∪ΓIr(R)
(2) We note that if we considerI=0, and R has a two-sided identity, then Γ0(R)=Γ(R).We say that a directed graphG 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 a and b in a graph G, the distance between a and b, denoted d(a,b), is the length of the shortest path from a to b if such a path exists; otherwise, d(a,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 ΓI(R) is always connected and its diameter, diam(ΓI(R)), is always less than or equal to 3 [5, Theorem 2.4]. The graph ΓI(R) is not in general strongly connected. For example, if we consider the case where R={(xy00)∣x,y∈ℤ2}, and I=0, then ΓI(R) is not strongly connected as a directed graph (see Figure 1). But we have the following theorem.Figure 1
ΓI(R), where R={(0x)0y∣x,y∈ℤ2} and I=0.Theorem 2.4.
LetI be an ideal of  R. If  Tl(I)=Tr(I), then  ΓI(R) is strongly connected with  diam(ΓI(R))≤3.Proof.
Letx and y be two distinct vertices of ΓI(R). Consider the following cases.(1)
IfxRy⊆I, then x→y is a path in ΓI(R).(2)
IfxRy⊈I, xRx⊆I and yRy⊆I, then, there exists r∈R with xry∈R∖I. In this case, x→xry→y is a path.(3)
IfxRy⊈I, xRx⊆I and yRy⊈I, then, there exists b∈R∖(I∪{x,y}) with bRy⊆I. If xRb⊆I, then x→b→y is a path. If xRb⊈I, there exists r∈R with xrb∉I. In this case, x→xrb→y is a path.(4)
IfxRy⊈I, xRx⊈I and yRy⊆I, then, there exists a∈R∖(I∪{x,y}) such that xRa⊆I. If aRy⊆I, then x→a→y is a path. If aRy⊈I, there exists r∈R with ary∉I. In this case, x→ary→y ia a path.(5)
IfxRy⊈I, xRx⊈I, and yRy⊈I, then, there exist a,b∈R∖(I∪{x,y}) with xRa⊆I and bRy⊆I. If a=b, then x→a→y is a path. If a≠b and aRb⊆I, then x→a→b→y is a path. If a≠b and aRb⊈I, there is r∈R with arb∉I; in this case, x→arb→y is a path.We now define an undirected graph as follows.Definition 2.5.
LetI be an ideal of R. We define an undirected graph Γ¯I(R) with vertices T(I)={x∈R∖I:xRy⊆IoryRx⊆Ifor somey∈R∖I}, where distinct vertices x and y are adjacent if and only if either xRy⊆I or yRx⊆I.Remark 2.6.
Note that the graphsΓI(R) and Γ¯I(R) share the same vertices and the same edges if the directions on the edges are ignored. Hence, the only difference between ΓI(R) and Γ¯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.
LetI be an ideal of  R. Then  Γ¯I(R) is connected with  diam(Γ¯I(R))≤3.Proof.
Letx,y∈Γ¯I(R) with x≠y. If xRy⊆I or yRx⊆I, then x-y is a path and d(x,y)=1. So assume that xRy⊈I and yRx⊈I. Consider the following cases.
Case 1.
xRx⊆I and yRy⊆I. As xRy⊈I, there exists r∈R with xry∈R∖I. In this case x-xry-y is a path and d(x,y)≤2.
Case 2.
xRx⊆I and yRy⊈I. There exists b∈R∖(I∪{x,y}) such that either bRy⊆I or yRb⊆I. If either bRx⊆I or xRb⊆I, then x-b-y is a path and d(x,y)≤2. Suppose that bRx⊈I and xRb⊈I. If yRb⊆I, then x-br1x-y is a path for some r1∈R for which br1x∈R∖I. If yRb⊆I, then x-xr2b-y is a path for some r2∈R for which xr2b∈R∖I. So in this case d(x,y)≤2.
Case 3.
IfxRx⊈I and yRy⊆I, a similar argument as in Case 2 shows that there exists a path of length at most 2 between x and y. SO d(x,y)≤2.
Case 4.
xRx⊈I and yRy⊈I. Then, there exist a,b∈R∖(I∪{x,y}) such that either aRx⊆I or xRa⊆I and such that either bRy⊆I or yRb⊆I. If a=b, then x-a-y is a path and d(x,y)≤2. If a≠b and either aRb⊆I or bRa⊆I, then x-a-b-y is a path and s(x,y)≤3. So assume that a≠b, aRb⊈I, and bRa⊈I. Then we have the following subcases.
Subcase 1.
IfaRy⊆I or yRa⊆I, then x-a-y is a path and d(x,y)≤2.
Subcase 2.
xRa⊆I and bRy⊆I. As aRb⊈I, there exists r∈R with arb∈R∖I. In this case, a-arb-y is a path and d(x,y)≤2.
Subcase 3.
aRx⊆I and yRb⊆I. As bRa⊈I, there exists r∈R with bra∈R∖I. Then x-bra-y is a path and d(x,y)≤2.
Subcase 4.
IfxRa⊆I, yRbI and aRy⊈I, there exists r∈R such that ary∈R∖I. In this case, x-ary-b-y is a path and d(x,y)≤3.
Subcase 5.
IfaRx⊆I, bRy⊆I and yRa⊈I, there exists r∈R with yra∈R∖I. Then x-yra-b-y is a path and d(x,y)≤3.
We have already shown that in any case, there exists a path betweenx and y and d(x,y)≤3. Thus diam(Γ¯I(R))≤3.As we mentioned in Figure1, if R is a noncommutative ring, the graph ΓI(R) need not be strongly connected as a directed graph, while as it is proved in Theorem 2.7, Γ¯I(R) is always connected.The girth of a graphG 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 ΓI(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 Γ¯I(R).Theorem 2.8.
LetI be an ideal of  R. If  Γ¯I(R) contains a cycle, then  gr(Γ¯I(R))≤4.Proof.
Suppose thatΓ¯I(R) contains a cycle x0-x1-x2-⋯-xn-1-xn-x0 of shortest length with n≥4 and look for a contradiction. Consider the following cases.
Case 1.
There exists1≤j≤n-1 such that xj∈A∶=Ixj-1-1∩Ixj+1-1. Without loss of generality, we may assume that j=1. If there exists y∈A∖I with y≠x1, then x0-x1-x2-y-x0 is a cycle in Γ¯I(R) which is a contradiction. So assume that A=I∪{x1}. Since x3-x4 is a path, either x3Rx4⊆I or x4Rx3⊆I. Note that x1Rx3⊈I, x3Rx1⊈I, x1Rx4⊈I, and x4Rx1⊈I. As A is an ideal of R and x1∈A, we have x3Rx1⊆A=I∪{x1}. There exists r∈R with x3rx1∈R∖I. But x3rx1∈A implies that x3rx1=x1. Similarly, one can shows that x4r′x1=x1 for some r′∈R. In this case, we have x3Rx1=x3Rx4r′x1⊆I which is a contradiction.
Case 2.
There exists1≤j≤n-1 with xj∈xj-1-1I∩xj+1-1I. A similar argument as in Case 1 leads us a contradiction.
Case 3.
For each1≤j≤n-1, xj∉Ixj-1-1∩Ixj+1-1, and xj∈xj-1-1I∩xj+1-1I. Therefore, without loss of generality, we can assume that we have a cycle in Γ¯I(R) of the form x0→x1→x2→⋯→xn-1→xn→x0 with all edges having only one direction. Now consider the following subcases.
Subcase 1.
x0Rx0⊆I and xnRxn⊆I. Since there is no directed path of the form x0→xn, there exists r∈R with x0rxn∈R∖I. In this case, x0→x0rxn→xn→x0 is a 3-cycle in Γ¯I(R) which is a contradiction.
Subcase 2.
x0Rx0⊆I and xnRxn⊈I. Since n≥4, x0Rxn-1⊈I. So there is r∈R with x0rxn-1∈R∖I. In this case, x0→x0rxn-1→xn→x0 is a 3-cycle in Γ¯I(R) which is a contradiction.
Subcase 3.
x0Rx0⊈I and xnRxn⊆I. Since n≥4, x1Rxn⊈I. So there is r∈R with x1rxn∈R∖I. In this case, x0→x1rxn→xn→x0 is a 3-cycle in Γ¯I(R) which is a contradiction.
Subcase 4.
x0Rx0⊈I and xnRxn⊈I. Since n≥4, x1Rxn-1⊈I. So there is r∈R with x1rxn-1∈R∖I. In this case, x0→x1rxn-1→xn→x0 is a 3-cycle in Γ¯I(R) which is a contradiction.
Since in each of these cases we have found a contradiction, we must havegr(Γ¯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]).
LetI be an ideal of R, and let x be an element of R. Set Ix-1={y∈R:yRx⊆I} and x-1I={y∈R:xRy⊆I}. Evidently both Ix-1 and x-1I are ideals of R containing I.Definition 3.2 (see [9]).
The elementx∈R is not right prime (nrp) (resp., not left prime (nlp)) to I if Ix-1≠I (resp., x-1I≠I). Otherwise, x isright prime (rp) (resp.,left prime (lp)) to I. Denote by Sr(I) (resp., Sl(I)) the set of elements of R that arenrp (resp.,nlp) to I. The ideal I of R is called a right primal (resp., left primal) ideal of R if Sr(I) (resp., Sl(I)) form an ideal of R, which is then termed the right (resp., left)adjoint ideal of I.Definition 3.3.
LetI be an ideal of R. The element x∈R isnot prime (np) to I if either Ix-1≠I or x-1I≠I.Otherwise, x is prime to I. Denote by S(I) the set of elements of R that arenp to I. Clearly S(I)=Sl(I)∪Sr(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]).
TakeR=ℤ2〈x,y〉, the noncommutative polynomial ring over ℤ2={0̅,1̅}, subject to xy=0,xRy=0. It is easy to check that y isnrp to the ideal (0), and Sr(0)=RyR. Moreover, for every z∈R,yRz=0 if and only if z=0. Hence y is nlp to (0). But as xRy=0 with y∉(0),x isnlp to (0). Therefore, Sl(0)=RxR. These show that (0) is both left and right primal ideal, but not a primal ideal.Remark 3.5.
(1) Note that ifR is commutative, then x being nrp to I is equivalent to x beingnlp and both are equivalent to x being not prime to I, and thus the definitions of Barnes [9] and Fuchs [8] are identical.
(2) IfR 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∶=Sr(I) (resp. P∶=Sl(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.
LetI be a proper ideal of  R. Then we have the following. (1)
AllSr(I), Sl(I), and S(I) contain I.(2)
ΓIl(R)=Sl(I)∖I. In particular, ΓIl(R)∪I=Sl(I).(3)
ΓIr(R)=Sr(I)∖I. In particular, ΓIr(R)∪I=Sr(I).(4)
ΓI(R)=S(I)∖I. In particular, ΓI(R)∪I=S(I).Proof.
(1) Assume thaty∈R∖I. For every x∈I, as xRy⊆I, we must have that x is nrp to I, that is x∈Sr(I). Hence I⊆Sr(I). Similarly, one can show that I⊆Sl(I) and I⊆S(I)
(2) Letx∈ΓIl(R). Then x∉I and xRy⊆I for some y∈R∖I. So x is nlp to I; hence x∈Sl(I)∖I. Thus ΓIl(R)⊆Sl(I)-I. For the other containment, assume that a∈Sl(I)-I. As a is nlp to I, there exists y∉I such that aRy⊆I. Thus a∈ΓIl(R), so Sl(I)-I⊆ΓIl(R). Therefore, we have the equality.
(3) The proof of(3) is similar to that of (2).
(4) By(2) and (3), we have ΓI(R)=ΓIl(R)∪ΓIr(R)=(Sl(I)∖I)∪(Sr(I)∖I)=(Sl(I)∪Sr(I))∖I=S(I)∖I.Theorem 3.7.
Assume thatR 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 ΓI(R)=∅,(2)
I is a right P-primal ideal of R if and only if I⊆P and ΓIr(R)=P∖I;(3)
I is a left P-primal ideal of R if and only if I⊆P and ΓIl(R)=P∖I;(4)
I is a P-primal ideal of R if and only if I⊆P and ΓI(R)=P∖I;Proof.
(1) Is clear.
(2) IfI is right P-primal, then P=Sr(I). Hence I⊆Sr(I)=P and ΓIr(R)=P∖I by Lemma 3.6. Conversely, if ΓIr(R)=P∖I, then P=ΓIr(R)∪I=Sr(I) by Lemma 3.6. Thus I is a right P-primal ideal of R.
The proofs of parts(3) and (4) are similar.LetI be an ideal of R. The prime radical of I, denoted by Rad(I), is the set of all a∈R such that the intersection of I with every m-system of R which contains a is nonempty. An ideal Q of R is a primary ideal if AB⊆Q and B⊈Q, where A and B are ideals of R, implies that A⊆Rad(Q). It was shown that an ideal Q of R is a primary ideal if and only if aRb⊆Q and b∉Q, where a,b∈R implies that a∈Rad(Q). The following theorem is another characterization via zero-divisor graphs.Theorem 3.8.
LetQ be an ideal of  R. Then  Q is a primary ideal of  R if and only if  ΓQ(R)=Rad(Q)∖Q.Proof.
IfQ is primary, then Q is a primal ideal of R with adjoint ideal Rad(Q). Thus ΓI(R)=Rad(Q)∖Q by Lemma 3.6. Conversely, assume that ΓI(R)=Rad(Q)∖Q and let aRb⊆Q for some a,b∈R. Assume that a,b∈R∖Q. Then a∈ΓI(R)=Rad(Q)∖Q, that is, Q is a primary ideal.



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