Domination number in the annihilating-submodule graph of modules over commutative rings

Let $M$ be a module over a commutative ring $R$. The annihilating-submodule graph of $M$, denoted by $AG(M)$, is a simple graph in which a non-zero submodule $N$ of $M$ is a vertex if and only if there exists a non-zero proper submodule $K$ of $M$ such that $NK=(0)$, where $NK$, the product of $N$ and $K$, is denoted by $(N:M)(K:M)M$ and two distinct vertices $N$ and $K$ are adjacent if and only if $NK=(0)$. This graph is a submodule version of the annihilating-ideal graph and under some conditions, is isomorphic with an induced subgraph of the Zariski topology-graph $G(\tau_T)$ which was introduced in (The Zariski topology-graph of modules over commutative rings, Comm. Algebra., 42 (2014), 3283--3296). In this paper, we study the domination number of $AG(M)$ and some connections between the graph-theoretic properties of $AG(M)$ and algebraic properties of module $M$.


Introduction
Throughout this paper R is a commutative ring with a non-zero identity and M is a unital R-module. By N ≤ M (resp. N < M ) we mean that N is a submodule (resp. proper submodule) of M . There are many papers on assigning graphs to rings or modules (see, for example, [4,7,12,13]). The annihilating-ideal graph AG(R) was introduced and studied in [13]. AG(R) is a graph whose vertices are ideals of R with nonzero annihilators and in which two vertices I and J are adjacent if and only if IJ = (0). Later, it was modified and further studied by many authors (see [1,2,3,18,20]).
In [7], the present authors introduced and studied the graph G(τ T ) (resp. AG(M )), called the Zariski topology-graph (resp. the annihilating-submodule graph), where T is a non-empty subset of Spec(M ). In this paper, we study the domination number of AG(M ) and some connections between the graph-theoretic properties of AG(M ) and algebraic properties of module M .
A prime submodule of M is a submodule P = M such that whenever re ∈ P for some r ∈ R and e ∈ M , we have r ∈ (P : M ) or e ∈ P [17].
The notations Z(R) and N il(R) will denote the set of all zero-divisors, the set of all nilpotent elements of R, respectively. Also, Z R (M ) or simply Z(M ), the set of zero divisors on M , is the set {r ∈ R| rm = 0 for some 0 = m ∈ M }. If Z(M ) = 0, then we say that M is a domain. An ideal I ≤ R is said to be nil if I consist of nilpotent elements.
Let us introduce some graphical notions and denotations that are used in what follows: A graph G is an ordered triple (V (G), E(G), ψ G ) consisting of a nonempty set of vertices, V (G), a set E(G) of edges, and an incident function ψ G that associates an unordered pair of distinct vertices with each edge. The edge e joins x and y if ψ G (e) = {x, y}, and we say x and y are adjacent. The number of edges incident at x in G is called the degree of the vertex x in G and is denoted by d G (v) or simply d(v). A path in graph G is a finite sequence of vertices {x 0 , x 1 , . . . , x n }, where x i−1 and x i are adjacent for each 1 ≤ i ≤ n and we denote x i−1 − x i for existing an edge between x i−1 and x i . The distance between two vertices x and y, denoted d(x, y), is the length of the shortest path from x to y. The diameter of a connected graph G is the maximum distance between two distinct vertices of G. For any vertex x of a connected graph G, the eccentricity of x, denoted e(x), is the maximum of the distances from x to the other vertices of G. The set of vertices with minimum eccentricity is called the center of the graph G, and this minimum eccentricity value is the radius of G. For some U ⊆ V (G), we denote by N (U ), the set of all vertices of G \ U adjacent to at least one vertex of U and A spanning subgraph H of G is called a perfect matching of G if every vertex of G has degree 1.
A clique of a graph is a complete subgraph and the supremum of the sizes of cliques in G, denoted by cl(G), is called the clique number of G. Let χ(G) denote the chromatic number of the graph G, that is, the minimal number of colors needed to color the vertices of G so that no two adjacent vertices have the same color. Obviously χ(G) ≥ cl(G).
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). A dominating set D is a connected dominating set if the subgraph < D > induced by D is a connected subgraph of G. The connected domination number of G, denoted by γ c (G), is the minimum cardinality of a connected dominating set of G. A dominating set D is a clique dominating set if the subgraph < D > induced by D is complete in G. The clique domination number γ cl (G) of G equals the minimum cardinality of a clique dominating set of G. A dominating set D is a paired-dominating set if the subgraph < D > induced by D has a perfect matching. The paired-domination number γ pr (G) of G equals the minimum cardinality of a paired-dominating set of G.
A Let S be a dominating set of a graph G, and u ∈ S. The private neighborhood of u relative to S in G is the set of vertices which are in the closed neighborhood of u, but not in the closed neighborhood of any vertex in S \ {u}. Thus the private neighborhood P N (u, S) of u with respect to S is given by has at least one private neighbor. An irredundant set S is a maximal irredundant set if for every vertex u ∈ V \S, the set S∪{u} is not irredundant. The irredundance number ir(G) is the minimum cardinality of maximal irredundant sets. There are so many domination parameters in the literature and for more details one can refer [15].
A bipartite graph is a graph whose vertices can be divided into two disjoint sets U and V such that every edge connects a vertex in U to one in V ; that is, U and V are each independent sets and complete bipartite graph on n and m vertices, denoted by K n,m , where V and U are of size n and m, respectively, and E(G) connects every vertex in V with all vertices in U . Note that a graph K 1,m is called a star graph and the vertex in the singleton partition is called the center of the graph. We denote by P n a path of order n (see [14]).
In section 2, a dominating set of AG(M ) is constructed using elements of the center when M is an Artinian module. Also we prove that the domination number of AG(M ) is equal to the number of factors in the Artinian decomposition of M and we also find several domination parameters of AG(M ). In section 3, we study the domination number of the annihilating-submodule graphs for reduced rings with finitely many minimal primes and faithful modules. Also, some relations between the domination numbers and the total domination numbers of annihilatingsubmodule graphs are studied.
The following results are useful for further reference in this paper.
Suppose that e is an idempotent element of R. We have the following statements. Proof. This is clear.
In the following result we find the total domination number of AG(M ). Proof. Let S be the set of all maximal elements of the set V (AG(M )), K ∈ S and |S| > 1. First we show that K = ann(annK) and there exists m ∈ M such that K = ann(m). Let K ∈ S. Then annK = 0 and so there exists 0 = m ∈ annK. Hence K ⊆ ann(annK) ⊆ ann(m). Thus by the maximality of K, we have K = ann(annK) = ann(m). By Zorn' Lemma it is clear that if V (AG(M )) = ∅, then S = ∅. For any K ∈ S choose m K ∈ M such that K = ann(m K ). We assert that D = {Rm K |K ∈ S} is a total dominating set of AG(M ). Since for every L ∈ V (AG(M )) there exists K ∈ S such that L ⊆ K = ann(m K ), L and Rm K are adjacent. Also for each pair K, K ′ ∈ S, we have (Rm K )(Rm K ′ ) = 0. Namely, if there exists m ∈ (Rm K )(Rm K ′ ) \ {0}, then K = K ′ = ann(m). Thus γ t ((AG(M ))) ≤ |S|. To complete the proof, we show that each element of an arbitrary γ t -set of AG(M ) is adjacent to exactly one element of S. Assume to the contrary, that a vertex L ′ of a γ t -set of AG(M ) is adjacent to K and K ′ , for K, K ′ ∈ S. Thus K = K ′ = annL ′ , which is impossible. Therefore γ t ((AG(M ))) = |S|. show that γ((AG(M ))) = n. Assume to the contrary, that B = {J 1 , . . . , J n−1 } is a dominating set for AG(M ). Since n ≥ 3, the submodules p i M and p j M , 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 M for i = 1, . . . , k, but none of the other of submodules from B are equal to some p s M (if B = {p 1 M, . . . , p n−1 M }, then p n M would be adjacent to some p i M , for i = n). So, every submodule in {p k+1 M, ..., p n M } is adjacent to a submodule in {J k+1 , ..., J n−1 }. It follows that for some s = t, there is an l such that (p s M )J l = 0 = (p t M )J l . Since p s p t , it follows that J l ⊆ p t M , so J 2 l = 0, which is impossible, since the ring R is reduced. So γ t ((AG(M ))) = γ((AG(M ))) = |M in(R)|.  (d) R has exactly two minimal primes.
Proof. Follows from Theorem 3.3 and Corollary 1.8.
In the following theorem the domination number of bipartite annihilating-submodule graphs is given. Proof. Let M be a faithful module. If AG(M ) is a bipartite graph, then from Theorem 1.7, either R is a reduced ring with exactly two minimal prime ideals, or AG(M ) is a star graph with more than one vertex. If R is a reduced ring with exactly two minimal prime ideals, then the result follows by Corollary 3.4. If AG(M ) is a star graph with more than one vertex, then we are done.
The next theorem is on the total domination number of the annihilating-submodule graphs of Artinian modules.
By Theorem 3.2, γ t ((AG(M ))) = |M ax(M )|. In the sequel, we prove that γ((AG(M ))) = n. Assume to the contrary, the set {K 1 , . . . , K n−1 } is a dominating set for AG(M ). Since M = M 1 ⊕ M 2 , where M 1 , M 2 are simple modules, we find that K i N s = K i N t = 0, for some i, t, s, where 1 ≤ i ≤ n − 1 and 1 ≤ t, s ≤ n. This means that K i = 0, a contradiction.
The following theorem provides an upper bound for the domination number of the annihilating-submodule graph of a Noetherian module. The remaining result of this paper provides the domination number of the annihilating-submodule graph of a finite direct product of modules. Proof. Parts (a) and (b) are trivial.