PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 97(111) (2015), 225–231 DOI: 10.2298/PIM140222001N

DOMINATION NUMBER IN THE ANNIHILATING-IDEAL GRAPHS OF COMMUTATIVE RINGS

Reza Nikandish, Hamid Reza Maimani, and Sima Kiani

Abstract. Let R be a commutative ring with identity and 𝔸(R) be the set of ideals with nonzero annihilator. The annihilating-ideal graph of R is defined as the graph 𝔸𝔾(R) with the vertex set 𝔸(R)^{∗} = 𝔸(R) ∖ {0} and two distinct vertices I and J are adjacent if and only if IJ = 0. In this paper, we study the domination number of 𝔸𝔾(R) and some connections between the domination numbers of annihilating-ideal graphs and zero-divisor graphs are given.

1. Introduction

The study of algebraic structures, using the properties of graphs, have become an exciting research topic in the past twenty years, leading to many interesting results and questions. There are many papers on assigning a graph to a ring, for instance see [1, 3, 9, 11].

Throughout this paper, all rings are assumed to be commutative rings with identity. By Min(R), Z(R) and Nil(R) we denote the set of all minimal prime ideals of R, the set of all zero-divisors of R and the set of all nilpotent elements of R, respectively. The socle of ring R, denoted by Soc(R), is the sum of all minimal ideals of R. If there are no minimal ideals, this sum is defined to be zero. A prime ideal 𝔭 is said to be an associated prime ideal of a commutative Noetherian ring R, if there exists a nonzero element x in R such that 𝔭 = ann(x). By Ass(R) we denote the set of all associated prime ideals of R. A ring R is said to be reduced, if it has no nonzero nilpotent element or equivalently ⋂ _{P ∈Min (R)} P = 0.

For every graph G, we denote by V (G), the vertex set of G. A bipartite graph is a graph all of whose vertices can be partitioned into two parts U and V such that every edge joins a vertex in U to one in V. A complete bipartite graph is a bipartite graph in which every vertex of one part is joined to every vertex of the other part. If one of the parts is a singleton, then the graph is said to be a star graph. A subset D of V (G) is called a dominating set if every vertex of G is either in D or adjacent to at least one vertex in D. The domination number of G, denoted by γ(G), is the number of vertices in a smallest dominating set of G. A total dominating set of a graph G is a set S of vertices of G such that every vertex is adjacent to a vertex in S. The total domination number of G, denoted by γ_{t}(G), is the minimum cardinality of a total dominating set. A dominating set of cardinality γ(G) (γ_{t}(G)) is called a γ-set (γ_{t}-set).

2010 Mathematics Subject Classification: Primary 13A15; Secondary 05C75.

Key words and phrases: annihilating-ideal graph, zero-divisor graph, domination number, minimal prime ideal.

The research of H. R. Maimani was partially supported by the grant 91050214 from IPM.

Communicated by Žarko Mijajlović.

Let R be a ring. The zero-divisor graph of R, Γ(R), is a graph with the vertex set Z(R) ∖ {0} and two distinct vertices x and y are adjacent if and only if xy = 0. The concept of the zero-divisor graph was first introduced by Beck (see [5]). We call an ideal I of R, an annihilating-ideal if there exists a nonzero ideal J of R such that IJ = 0. We use the notation 𝔸(R) for the set of all annihilating-ideals of R. By the annihilating-ideal graph of R, 𝔸𝔾(R), we mean the graph with the vertex set 𝔸(R)^{∗} = 𝔸(R) ∖ {0} such that two distinct vertices I and J are adjacent if and only if IJ = 0. The annihilating-ideal graph was first introduced in [6] and some interesting properties of this graph have been studied. In this article, we study the domination number of the annihilating-ideal graphs. For reduced rings with finitely many minimal primes and Artinian rings, the domination number of the annihilating-ideal graphs is given. Also, some relations between the domination numbers of annihilating-ideal graphs and zero-divisor graphs are studied.

2. Main results

We start with the following remark which completely characterizes all rings for which either γ(𝔸𝔾(R)) = 1 or γ(Γ(R)) = 1.

Remark 2.1. Let R be a ring. By [6, Theorem 2.2], there is a vertex of 𝔸𝔾(R) which is adjacent to every other vertex if and only if either R ≅ F × D, where F is a field and D is an integral domain, or Z(R) is an annihilator ideal. Also, by [3, Theorem 2.5], there is a vertex of Γ(R) which is adjacent to every other vertex if and only if either R ≅ ℤ_{2} × D, where D is an integral domain, or Z(R) is an annihilator ideal. Now, let R be a reduced ring. Then γ(𝔸𝔾(R)) = 1 if and only if R ≅ F ×D, where F is a field and D is an integral domain. Moreover, γ(Γ(R)) = 1 if and only if R ≅ ℤ_{2} × D, where D is an integral domain.

Now, we can state the following proposition.

Proposition 2.1. Let R be a ring. Then we have the following:

(i) If γ(𝔸𝔾(R)) = 1, then γ(Γ(R)) ∈ {1, 2}. (ii) If γ(Γ(R)) = 1, then γ(𝔸𝔾(R)) = 1.

Proof. The result follows from Remark 2.1 and [10, Proposition 8]. □

The following result describes the relation between γ_{t}(𝔸𝔾(R)) (γ_{t}(Γ(R))) and γ(𝔸𝔾(R)) (γ(Γ(R))).

Theorem 2.1. Let R be a ring. Then

(i) γ_{t}(𝔸𝔾(R)) = γ(𝔸𝔾(R)) or γ_{t}(𝔸𝔾(R)) = γ(𝔸𝔾(R)) + 1.

(ii) γ_{t}(Γ(R)) = γ(Γ(R)) or γ_{t}(Γ(R)) = γ(Γ(R)) + 1.

Proof. (i) Let γ_{t}(𝔸𝔾(R)) ≠ γ(𝔸𝔾(R)) and D be a γ-set of 𝔸𝔾(R). If γ(𝔸𝔾(R)) = 1, then it is clear that γ_{t}(𝔸𝔾(R)) = 2. So let γ(𝔸𝔾(R)) > 1 and put k = Max{n | ∃ I_{1}, …, I_{n} ∈ D s.t. ∏ _{i=1}^{n} I_{i} ≠ 0}. Since γ_{t}(𝔸𝔾(R)) ≠ γ(𝔸𝔾(R)), we have k ⩾ 2. Let I_{1}, …, I_{k} ∈ D be such that ∏ _{i=1}^{k} I_{i} ≠ 0. Then S = {∏ _{i=1}^{k} I_{i}, annI_{1}, …, annI_{k}}∪D ∖{I_{1}, …, I_{k}} is a γ_{t}-set. Hence γ_{t}(𝔸𝔾(R)) = γ(𝔸𝔾(R)) + 1.

(ii) It is clear by the same argument of part (i). □

In the following result we find the total domination number of 𝔸𝔾(R).

Theorem 2.2. Let M be the set of all maximal elements of the set 𝔸(R). If |M| > 1, then γ_{t}(𝔸𝔾(R)) = |M|.

Proof. Let M be the set of all maximal elements of the set 𝔸(R), I ∈ M and |M| > 1. First we show that I = ann(annI) and there exists x ∈ R such that I = ann(x). Let I ∈ M. Then annI ≠ 0 and so there exists 0 ≠ x ∈ annI. Hence I ⊆ ann(annI) ⊆ ann(x). Thus by the maximality of I, we have I = ann(annI) = ann(x). By Zorn’s Lemma it is clear that if 𝔸(R) ≠ ∅, then M ≠ ∅. For any I ∈ M choose x_{I} ∈ R such that I = ann(x_{I}). We assert that D = {Rx_{I} | I ∈ M} is a total dominating set of 𝔸𝔾(R). Since for every J ∈ 𝔸(R) there exists I ∈ M such that J ⊆ I = ann(x_{I}), J and Rx_{I} are adjacent. Also for each pair I, I′ ∈ M, we have Rx_{I} Rx_{I′} = 0. Namely, if there exists x ∈ Rx_{I} Rx_{I′} ∖ {0}, then I = I′ = ann(x). Thus γ_{t}(𝔸𝔾(R)) ⩽ |M|. To complete the proof, we show that each element of an arbitrary γ_{t}-set of 𝔸𝔾(R) is adjacent to exactly one element of M. Assume to the contrary, that a vertex K of a γ_{t}-set of 𝔸𝔾(R) is adjacent to I and I′, for I, I′ ∈ M. Thus I = I′ = annK, which is impossible. Therefore γ_{t}(𝔸𝔾(R)) = |M|. □

Theorem 2.3. Let R be a ring. Then γ_{t}(Γ(R)) ⩽ γ_{t}(𝔸𝔾(R)).

Proof. Let γ_{t}(𝔸𝔾(R)) = k. In the light of the proof of Theorem 2.2, there exists x_{i} ∈ R such that ann(x_{i}) is a maximal ideal of 𝔸(R), for i = 1, …, k, and D = {Rx_{1}, …, Rx_{k}} is a minimum total dominating set of 𝔸𝔾(R). It is easy to check that D′ = {x_{1}, …, x_{k}} is a total dominating set of Γ(R). Thus γ_{t}(Γ(R)) ⩽ γ_{t}(𝔸𝔾(R)). □

It is interesting to find some rings for which γ_{t}(Γ(R)) = γ(Γ(R)) = γ_{t}(𝔸𝔾(R)) = γ(𝔸𝔾(R)). In the next result, we study the domination number of the annihilating-ideal graphs of reduced rings with finitely many minimal primes.

Theorem 2.4. Let R be a reduced ring and | Min(R)| < ∞. If γ(𝔸𝔾(R)) > 1, then γ(𝔸𝔾(R)) = γ_{t}(𝔸𝔾(R)) = | Min(R)|.

Proof. Since R is reduced and γ(𝔸𝔾(R)) > 1, we have | Min(R)| > 1. Suppose that Min(R) = {𝔭_{1}, …, 𝔭_{n}}. If n = 2, the result follows from [7, Corollary 2.5]. Therefore, suppose that n ⩾ 3. Define 𝔭̂_{i} = 𝔭_{1} …𝔭_{i−1} 𝔭_{i+1} …𝔭_{n}, for every i = 1, …, n. Clearly, 𝔭̂_{i} ≠ 0, for every i = 1, …, n. Since R is reduced, we deduce that 𝔭̂_{i}𝔭_{i} = 0. Therefore, every 𝔭_{i} is a vertex of 𝔸𝔾(R). If I is a vertex of 𝔸𝔾(R), then by [8, Corollary 2.4], I ⊆ Z(R) = ⋃ _{i=1}^{n} 𝔭_{i}. It follows from the Prime Avoidance Theorem (see [12, Theorem 3.61]) that I ⊆ 𝔭_{i}, for some i, 1 ⩽ i ⩽ n. Thus 𝔭_{i} is a maximal element of 𝔸(R), for every i = 1, …, n. From Theorem 2.2, γ_{t}(𝔸𝔾(R)) = | Min(R)|. Now, we show that γ(𝔸𝔾(R)) = n. Assume to the contrary, that B = {J_{1}, …, J_{n−1} } is a dominating set for 𝔸𝔾(R). Since n ⩾ 3, the ideals 𝔭_{i} and 𝔭_{j}, for i ≠ j are not adjacent (from 𝔭_{i}𝔭_{j} = 0 ⊆ 𝔭_{k} it would follow that 𝔭_{i} ⊆ 𝔭_{k}, or 𝔭_{j} ⊆ 𝔭_{k} which is not true). Because of that, we may assume that for some k < n − 1, J_{i} = 𝔭_{i} for i = \overline{1, k}, but none of the other of ideals from B are equal to some 𝔭_{s} (if B = {𝔭_{1}, …, 𝔭_{n−1} } then 𝔭_{n} would be adjacent to some 𝔭_{i}, for i ≠ n). So, every ideal in {𝔭_{k+1} , …, 𝔭_{n}} is adjacent to an ideal in {J_{k+1} , …, J_{n−1} }. It follows that for some s ≠ t there is an l such that 𝔭_{s}J_{l} = 0 = 𝔭_{t}J_{l}. Since 𝔭_{s} ⊈ 𝔭_{t}, it follows that J_{l} ⊂ 𝔭_{t}, so J_{l}^{2} = 0, which is impossible, since the ring R is reduced. So γ(𝔸𝔾(R)) = γ_{t}(𝔸𝔾(R)) = | Min(R)|. □

Theorem 2.4 leads to the following corollary.

Corollary 2.1. Let R be a reduced ring. If γ(𝔸𝔾(R)) > 1, then the following are equivalent:

(i) γ(𝔸𝔾(R)) = 2.

(ii) 𝔸𝔾(R) is a bipartite graph with two nonempty parts.

(iii) 𝔸𝔾(R) is a complete bipartite graph with two nonempty parts.

(iv) R has exactly two minimal primes.

Proof. The result follows from Theorem 2.4 and [7, Corollary 2.5]. □

In Theorem 15 of [10], it is proved that if R is a finite reduced ring such that γ(Γ(R)) ≠ 1, then γ(Γ(R)) = | Min(R)|. In the next theorem, we prove this result, where R is not necessarily finite.

Theorem 2.5. Let R be a reduced ring and | Min(R)| < ∞. If γ(Γ(R)) > 1, then γ(Γ(R)) = γ_{t}(Γ(R)) = | Min(R)|.

Proof. Using the notations in the proof of Theorem 2.4, set A = {x̂_{i} | 1 ⩽ i ⩽ n}, where, for every i, x̂_{i} is an element of 𝔭̂_{i}. We show that A is a dominating set in Γ(R). Since R is reduced, it is easily seen that x̂_{i} is a vertex of Γ(R), for i = 1, …, n. Assume that x ∉ A is a vertex of Γ(R). Then x ∈ 𝔭_{i}, for some i. The equality x̂_{i}𝔭_{i} = 0 implies that xx̂_{i} = 0. In the sequel, we prove that γ(Γ(R)) = n. If n = 2, then [2, Theorem 2.4] completes the proof. Thus assume that n ⩾ 3. Assume to the contrary, the set B = {y_{1}, …, y_{n−1} } is a dominating set for Γ(R). By the Prime Avoidance Theorem, there exists x_{i} ∈ 𝔭_{i} ∖ ⋃ _{j=1,j≠i}^{n} 𝔭_{j}. Thus there exists k, 1 ⩽ k ⩽ n − 1, such that y_{k}x_{i} = y_{k}x_{j} = 0, for some different i, j, 1 ⩽ i, j ⩽ n. Since x_{i} ∉ 𝔭_{j} and x_{j} ∉ 𝔭_{i}, we have y_{k} ∈ 𝔭_{i} ∩ 𝔭_{j}. As R is a reduced ring, we conclude that y_{k} ∉ 𝔭_{l}, for some l, 1 ⩽ l ⩽ n. Now, y_{k}x_{i} = 0 ∈ 𝔭_{l} implies that either y_{k} ∈ 𝔭_{l} or x_{i} ∈ 𝔭_{l}, a contradiction. Clearly, x̂_{i} x̂_{j} = 0, where i ≠ j. Therefore, γ(Γ(R)) = γ_{t}(Γ(R)) = | Min(R)|. □

Corollary 2.2. Let R be a reduced ring and | Min(R)| < ∞. If γ(𝔸𝔾(R)) ≠ 1, then γ(𝔸𝔾(R)) = γ(Γ(R)).

Proof. The result follows from Part (2) of Proposition 2.1 and Theorems 2.4 and 2.5. □

In the following theorem the domination number of bipartite annihilating-ideal graphs is given.

Theorem 2.6. If 𝔸𝔾(R) is a bipartite graph, then γ(𝔸𝔾(R)) ⩽ 2.

Proof. If 𝔸𝔾(R) is bipartite, it follows from [1, Theorem 27] that one of the following cases occurs: (a): 𝔸𝔾(R) is a star graph; (b): 𝔸𝔾(R) is the path of order 4; (c): Nil(R) = Soc(R). If (a) or (b) happen, then we are done. Suppose that Nil(R) = Soc(R). If R is reduced, then the result follows from Corollary 2.1. If R is nonreduced, then [1, Theorem 20] completes the proof. □

The next theorem is on the domination number of the annihilating-ideal graphs of Artinian rings.

Theorem 2.7. Let R be an Artinian ring and R ≇ F_{1} × F_{2}, where F_{1} and F_{2} are two fields. Then γ(𝔸𝔾(R)) = γ_{t}(𝔸𝔾(R)) = | Min(R)|.

Proof. Since R is Artinian, we deduce that each ideal of R is an annihilatingideal. So, the set of maximal elements of 𝔸(R) and Max(R) are equal. By [4, Theorem 8.7], R ≅ R_{1} × ⋯× R_{k}, where (R_{i}, 𝔪_{i}) is an Artinian local ring, for i = 1, …, k. Let Max(R) = {𝔫_{i} = R_{1} × ⋯× R_{i−1} × 𝔪_{i} × R_{i+1} × ⋯× R_{k} | 1 ⩽ i ⩽ k}. By Theorem 2.2, γ_{t}(𝔸𝔾(R)) = | Max(R)|. In the sequel, we prove that γ(𝔸𝔾(R)) = k. Assume to the contrary, the set {J_{1}, …, J_{k−1} } is a dominating set for 𝔸𝔾(R). Since R ≇ F_{1} × F_{2}, where F_{1} and F_{2} are two fields, we find that J_{i}𝔫_{s} = J_{i}𝔫_{t} = 0, for some i, t, s, where 1 ⩽ i ⩽ k − 1 and 1 ⩽ t, s ⩽ k. This means that J_{i} = 0, a contradiction. □

The condition of R to be an Artinian ring in the previous theorem is necessary; see the next example.

Example 2.1. Let R = \frac{k[x,y,z]}{(xy,xz,yz)} , where k is a field and x, y and z are indeterminates. Then γ(𝔸𝔾(R)) = 2 but | Min(R)| > 2.

Theorem 2.7 gives the following immediate corollary.

Corollary 2.3. Let R be an Artinian ring and R ≇ F_{1} × F_{2}, where F_{1} and F_{2} are two fields. Then γ(𝔸𝔾(R)) = γ(Γ(R)) = | Min(R)|.

Proof. The result follows from Theorem 2.7 and [10, Theorem 11]. □

Example 2.2. Let n be a natural number and n = p_{1}^{n_{1}} p_{2}^{n_{2}} …p_{m}^{n_{m}} , where p_{i}’s are distinct primes and n_{i}’s are natural numbers. Then one of the following holds:

(i) γ(𝔸𝔾(ℤ_{n})) = 1 if and only if either n = p_{1}p_{2} or n = p_{1}^{n_{1}} , where n_{1} > 1. (ii) γ(𝔸𝔾(ℤ_{n})) = 2 if and only if m = 2 and either n_{1} > 1 or n_{2} > 1. (iii) If m ⩾ 3, then γ(𝔸𝔾(ℤ_{n})) = m.

The following theorem provides an upper bound for the domination number of the annihilating-ideal graph of a Noetherian ring.

Theorem 2.8. If R is a Notherian ring, then γ(𝔸𝔾(R)) ⩽ | Ass(R)| < ∞.

Proof. By [12, Remark 9.33], | Ass(R)| < ∞. Let Ass(R) = {𝔭_{1}, …, 𝔭_{n}} and 𝔭_{i} = ann(x_{i}), for some x_{i} ∈ R and for every i = 1, …, n. Set A = {Rx_{i} | 1 ⩽ i ⩽ n}. We show that A is a dominating set of 𝔸𝔾(R). Clearly, every Rx_{i} is a vertex of 𝔸𝔾(R), for i = 1, …, n. If I is a vertex of 𝔸𝔾(R), then [12, Corollary 9.36] implies that I ⊆ Z(R) = ⋃ _{i=1}^{n} 𝔭_{i}. It follows from the Prime Avoidance Theorem that I ⊆ 𝔭_{i}, for some i, 1 ⩽ i ⩽ n. Thus IRx_{i} = 0, as desired. □

The inequality in the above theorem may be strict; see the next example.

Example 2.3. Let R = \frac{k[x,y]}{(x^{2} ,xy)} , where k is a field and x, y are indeterminate. Then Ass(R) = {(x), (x, y)} but γ(𝔸𝔾(R)) = 1 (Indeed the vertex (x) is adjacent to all other vertices).

Remark 2.2. If R is Noetherian, then by a similar argument to that of the proof of Theorem 2.8, one can show that γ(Γ(R)) ⩽ | Ass(R)|.

We end the paper with the following result about the domination number of the annihilating-ideal graph of a finite direct product of rings.

Theorem 2.9. For a ring R, which is a product of two (nonzero) rings, one of the following holds:

(i) If R ≅ F × D, where F is a field and D is an integral domain, then γ(𝔸𝔾(R)) = 1.

(ii) If R ≅ D_{1} × D_{2}, where D_{1} and D_{2} are integral domains which are not fields, then γ(𝔸𝔾(R)) = 2.

(iii) If R ≅ R_{1} × D, where R_{1} is a ring which is not integral domain and D is an integral domain, then γ(𝔸𝔾(R)) = γ(𝔸𝔾(R_{1})) + 1.

(iv) If R ≅ R_{1} × R_{2}, where R_{1} and R_{2} are two rings which are not integral domains, then γ(𝔸𝔾(R)) = γ(𝔸𝔾(R_{1})) + γ(𝔸𝔾(R_{2})).

Proof. Parts (i) and (ii) are clear.

(iii) With no loss of generality, one can assume that γ(𝔸𝔾(R_{1})) < ∞. Suppose that γ(𝔸𝔾(R_{1})) = n and {I_{1}, …, I_{n}} is a minimal dominating set of 𝔸𝔾(R_{1}). It is not hard to see that {I_{1} × 0, …, I_{n} × 0, 0 × D} is the smallest dominating set of 𝔸𝔾(R).

(iv) We may assume that γ(𝔸𝔾(R_{1})) = m and γ(𝔸𝔾(R_{2})) = n, for some positive integers m and n. Let {I_{1}, …, I_{m}} and {J_{1}, …, J_{n}} be two minimal dominating sets in 𝔸𝔾(R_{1}) and 𝔸𝔾(R_{1}), respectively. It is easily seen that {I_{1} × 0, …, I_{m} × 0, 0 × J_{1} …0 × J_{n}} is the smallest dominating set in 𝔸𝔾(R). □

Acknowledgements. The authors are deeply grateful to the referee for his/her valuable suggestions.

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Department of Mathematics (Received 01 05 2013) Jundi-Shapur University of Technology (Revised 01 01 2014) Dezful, Iran

r.nikandish@jsu.ac.ir

Mathematics Section Department of Basic Sciences Shahid Rajaee Teacher Training University Tehran, Iran and School of Mathematics Institute for Research in Fundamental Sciences (IPM) Tehran, Iran

maimani@ipm.ir

Department of Mathematics Science and Research Branch Islamic Azad University Tehran, Iran

s.kiani@srbiau.ac.ir