Zero-Divisor Graphs with respect to Ideals in Noncommutative Rings

Abstract

Let 𝑅 be a commutative ring and 𝐼 an ideal of 𝑅. The zero-divisor graph of 𝑅 with respect to 𝐼, denoted Γ_𝐼(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.

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 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 𝑆_𝑟(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_𝑟(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 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 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 Γ^𝑙𝐼(𝑅) with vertices 𝑇^𝑙(𝐼)={𝑥∈𝑅⧵𝐼∶𝑥𝑅𝑦⊆𝐼forsome𝑦∈𝑅⧵𝐼}, and where 𝑥→𝑦 is a directed edge between distinct vertices 𝑥 and 𝑦 if and only if 𝑥𝑅𝑦⊆𝐼. The set 𝑇^𝑟(𝐼) and the graph Γ^𝑟𝐼(𝑅) are defined in a similar way.

Definition 2.2. Let 𝐼 be an ideal of 𝑅. We define a directed graph Γ_𝐼(𝑅) with vertices 𝑇(𝐼)={𝑥∈𝑅⧵𝐼∶𝑥𝑅𝑦⊆𝐼or𝑦𝑅𝑥⊆𝐼forsome𝑦∈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 graphs 𝐺₁ 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)=𝑇^𝑙(𝐼)∪𝑇^𝑟(𝐼) and 𝐸(Γ_𝐼(𝑅))=𝐸(Γ^𝑙𝐼(𝑅))∪𝐸(Γ^𝑟𝐼(𝑅)). Therefore, Γ_𝐼(𝑅)=Γ^𝑙𝐼(𝑅)∪Γ^𝑟𝐼(𝑅)
(2) We note that if we consider 𝐼=0, and 𝑅 has a two-sided identity, then Γ₀(𝑅)=Γ(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 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 Γ_𝐼(𝑅) is always connected and its diameter, diam(Γ_𝐼(𝑅)), is always less than or equal to 3 [5, Theorem 2.4]. The graph Γ_𝐼(𝑅) is not in general strongly connected. For example, if we consider the case where 𝑅={^𝑥𝑦00∣𝑥,𝑦∈ℤ₂}, and 𝐼=0, then Γ_𝐼(𝑅) is not strongly connected as a directed graph (see Figure 1). But we have the following theorem.

Figure 1 

Γ𝐼(𝑅), where 𝑅={(𝑥0𝑦0)∣𝑥,𝑦∈ℤ2} and 𝐼=0.

Theorem 2.4. Let 𝐼 be an ideal of 𝑅. If 𝑇^𝑙(𝐼)=𝑇^𝑟(𝐼), then Γ_𝐼(𝑅) is strongly connected with diam(Γ_𝐼(𝑅))≤3.

Proof. Let 𝑥 and 𝑦 be two distinct vertices of Γ_𝐼(𝑅). Consider the following cases.(1)If 𝑥𝑅𝑦⊆𝐼, then 𝑥→𝑦 is a path in Γ_𝐼(𝑅).(2)If 𝑥𝑅𝑦⊈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)If 𝑥𝑅𝑦⊈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)If 𝑥𝑅𝑦⊈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)If 𝑥𝑅𝑦⊈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. Let 𝐼 be an ideal of 𝑅. We define an undirected graph Γ_𝐼(𝑅) with vertices 𝑇(𝐼)={𝑥∈𝑅⧵𝐼∶𝑥𝑅𝑦⊆𝐼or𝑦𝑅𝑥⊆𝐼forsome𝑦∈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 graphs Γ_𝐼(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 Γ_𝐼(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. Let 𝐼 be an ideal of 𝑅. Then Γ_𝐼(𝑅) is connected with diam(Γ_𝐼(𝑅))≤3.

Proof. Let 𝑥,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. If 𝑥𝑅𝑥⊈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. If 𝑎𝑅𝑦⊆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. If 𝑥𝑅𝑎⊆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. If 𝑎𝑅𝑥⊆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 between 𝑥 and 𝑦 and 𝑑(𝑥,𝑦)≤3. Thus diam(Γ_𝐼(𝑅))≤3.

As we mentioned in Figure 1, if 𝑅 is a noncommutative ring, the graph Γ_𝐼(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 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 Γ_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 Γ_𝐼(𝑅).

Theorem 2.8. Let 𝐼 be an ideal of 𝑅. If Γ_𝐼(𝑅) contains a cycle, then 𝑔𝑟(Γ_𝐼(𝑅))≤4.

Proof. Suppose that Γ_𝐼(𝑅) contains a cycle 𝑥₀−𝑥₁−𝑥₂−⋯−𝑥_𝑛−1−𝑥_𝑛−𝑥₀ of shortest length with 𝑛≥4 and look for a contradiction. Consider the following cases.
Case 1. There exists 1≤𝑗≤𝑛−1 such that 𝑥_𝑗∈𝐴∶=𝐼𝑥^−1𝑗−1∩𝐼𝑥^−1𝑗+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 exists 1≤𝑗≤𝑛−1 with 𝑥_𝑗∈𝑥^−1𝑗−1𝐼∩𝑥^−1𝑗+1𝐼. A similar argument as in Case 1 leads us a contradiction.
Case 3. For each 1≤𝑗≤𝑛−1, 𝑥_𝑗∉𝐼𝑥^−1𝑗−1∩𝐼𝑥^−1𝑗+1, and 𝑥_𝑗∈𝑥^−1𝑗−1𝐼∩𝑥^−1𝑗+1𝐼. Therefore, without loss of generality, we can assume that we have a cycle in Γ_𝐼(𝑅) of the form 𝑥₀→𝑥₁→𝑥₂→⋯→𝑥_𝑛−1→𝑥_𝑛→𝑥₀ with all edges having only one direction. Now consider the following subcases.
Subcase 1. 𝑥₀𝑅𝑥₀⊆𝐼 and 𝑥_𝑛𝑅𝑥_𝑛⊆𝐼. Since there is no directed path of the form 𝑥₀→𝑥_𝑛, there exists 𝑟∈𝑅 with 𝑥₀𝑟𝑥_𝑛∈𝑅⧵𝐼. In this case, 𝑥₀→𝑥₀𝑟𝑥_𝑛→𝑥_𝑛→𝑥₀ is a 3-cycle in Γ_𝐼(𝑅) which is a contradiction.
Subcase 2. 𝑥₀𝑅𝑥₀⊆𝐼 and 𝑥_𝑛𝑅𝑥_𝑛⊈𝐼. Since 𝑛≥4, 𝑥₀𝑅𝑥_𝑛−1⊈𝐼. So there is 𝑟∈𝑅 with 𝑥₀𝑟𝑥_𝑛−1∈𝑅⧵𝐼. In this case, 𝑥₀→𝑥₀𝑟𝑥_𝑛−1→𝑥_𝑛→𝑥₀ is a 3-cycle in Γ_𝐼(𝑅) which is a contradiction.
Subcase 3. 𝑥₀𝑅𝑥₀⊈𝐼 and 𝑥_𝑛𝑅𝑥_𝑛⊆𝐼. Since 𝑛≥4, 𝑥₁𝑅𝑥_𝑛⊈𝐼. So there is 𝑟∈𝑅 with 𝑥₁𝑟𝑥_𝑛∈𝑅⧵𝐼. In this case, 𝑥₀→𝑥₁𝑟𝑥_𝑛→𝑥_𝑛→𝑥₀ is a 3-cycle in Γ_𝐼(𝑅) which is a contradiction.
Subcase 4. 𝑥₀𝑅𝑥₀⊈𝐼 and 𝑥_𝑛𝑅𝑥_𝑛⊈𝐼. Since 𝑛≥4, 𝑥₁𝑅𝑥_𝑛−1⊈𝐼. So there is 𝑟∈𝑅 with 𝑥₁𝑟𝑥_𝑛−1∈𝑅⧵𝐼. In this case, 𝑥₀→𝑥₁𝑟𝑥_𝑛−1→𝑥_𝑛→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 have 𝑔𝑟(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]). Let 𝐼 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 element 𝑥∈𝑅 is not right prime (nrp) (resp., not left prime (nlp)) to 𝐼 if 𝐼𝑥⁻¹≠𝐼 (resp., 𝑥⁻¹𝐼≠𝐼). Otherwise, 𝑥 is right prime (rp) (resp., left prime (lp)) to 𝐼. Denote by 𝑆_𝑟(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_𝑟(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 𝑥∈𝑅 is not 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 are np to 𝐼. Clearly 𝑆(𝐼)=𝑆_𝑙(𝐼)∪𝑆_𝑟(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=ℤ₂⟨𝑥,𝑦⟩, the noncommutative polynomial ring over ℤ₂={0,1}, subject to 𝑥𝑦=0, 𝑥𝑅𝑦=0. It is easy to check that 𝑦 is nrp to the ideal (0), and 𝑆_𝑟(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 is nlp to (0). Therefore, 𝑆_𝑙(0)=𝑅𝑥𝑅. 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 𝑥 being nrp to 𝐼 is equivalent to 𝑥 being nlp and both are equivalent to 𝑥 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_𝑟(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_𝑟(𝐼), 𝑆_𝑙(𝐼), and 𝑆(𝐼) contain 𝐼.(2)Γ^𝑙𝐼(𝑅)=𝑆_𝑙(𝐼)⧵𝐼. In particular, Γ^𝑙𝐼(𝑅)∪𝐼=𝑆_𝑙(𝐼).(3)Γ^𝑟𝐼(𝑅)=𝑆_𝑟(𝐼)⧵𝐼. In particular, Γ^𝑟𝐼(𝑅)∪𝐼=𝑆_𝑟(𝐼).(4)Γ_𝐼(𝑅)=𝑆(𝐼)⧵𝐼. In particular, Γ_𝐼(𝑅)∪𝐼=𝑆(𝐼).

Proof. (1) Assume that 𝑦∈𝑅⧵𝐼. For every 𝑥∈𝐼, as 𝑥𝑅𝑦⊆𝐼, we must have that 𝑥 is nrp to 𝐼, that is 𝑥∈𝑆_𝑟(𝐼). Hence 𝐼⊆𝑆_𝑟(𝐼). Similarly, one can show that 𝐼⊆𝑆_𝑙(𝐼) and 𝐼⊆𝑆(𝐼)
(2) Let 𝑥∈Γ^𝑙𝐼(𝑅). Then 𝑥∉𝐼 and 𝑥𝑅𝑦⊆𝐼 for some 𝑦∈𝑅⧵𝐼. So 𝑥 is nlp to 𝐼; hence 𝑥∈𝑆_𝑙(𝐼)⧵𝐼. Thus Γ^𝑙𝐼(𝑅)⊆𝑆_𝑙(𝐼)−𝐼. For the other containment, assume that 𝑎∈𝑆_𝑙(𝐼)−𝐼. As 𝑎 is nlp to 𝐼, there exists 𝑦∉𝐼 such that 𝑎𝑅𝑦⊆𝐼. Thus 𝑎∈Γ^𝑙𝐼(𝑅), so 𝑆_𝑙(𝐼)−𝐼⊆Γ^𝑙𝐼(𝑅). Therefore, we have the equality.
(3) The proof of (3) is similar to that of (2).
(4) By (2) and (3), we have Γ_𝐼(𝑅)=Γ^𝑙𝐼(𝑅)∪Γ^𝑟𝐼(𝑅)=(𝑆_𝑙(𝐼)⧵𝐼)∪(𝑆_𝑟(𝐼)⧵𝐼)=(𝑆_𝑙(𝐼)∪𝑆_𝑟(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 Γ_𝐼(𝑅)=∅,(2)𝐼 is a right 𝑃-primal ideal of 𝑅 if and only if 𝐼⊆𝑃 and Γ^𝑟𝐼(𝑅)=𝑃⧵𝐼;(3)𝐼 is a left 𝑃-primal ideal of 𝑅 if and only if 𝐼⊆𝑃 and Γ^𝑙𝐼(𝑅)=𝑃⧵𝐼;(4)𝐼 is a 𝑃-primal ideal of 𝑅 if and only if 𝐼⊆𝑃 and Γ_𝐼(𝑅)=𝑃⧵𝐼;

Proof. (1) Is clear.
(2) If 𝐼 is right 𝑃-primal, then 𝑃=𝑆_𝑟(𝐼). Hence 𝐼⊆𝑆_𝑟(𝐼)=𝑃 and Γ^𝑟𝐼(𝑅)=𝑃⧵𝐼 by Lemma 3.6. Conversely, if Γ^𝑟𝐼(𝑅)=𝑃⧵𝐼, then 𝑃=Γ^𝑟𝐼(𝑅)∪𝐼=𝑆_𝑟(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 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. Let 𝑄 be an ideal of 𝑅. Then 𝑄 is a primary ideal of 𝑅 if and only if Γ_𝑄(𝑅)=Rad(𝑄)⧵𝑄.

Proof. If 𝑄 is primary, then 𝑄 is a primal ideal of 𝑅 with adjoint ideal Rad(𝑄). Thus Γ_𝐼(𝑅)=Rad(𝑄)⧵𝑄 by Lemma 3.6. Conversely, assume that Γ_𝐼(𝑅)=Rad(𝑄)⧵𝑄 and let 𝑎𝑅𝑏⊆𝑄 for some 𝑎,𝑏∈𝑅. Assume that 𝑎,𝑏∈𝑅⧵𝑄. Then 𝑎∈Γ_𝐼(𝑅)=Rad(𝑄)⧵𝑄, that is, 𝑄 is a primary ideal.

Acknowledgment

The authors thank the referee for valuable comments.