DOMINATION NUMBER IN THE ANNIHILATING-SUBMODULE GRAPH OF MODULES OVER COMMUTATIVE RINGS

Abstract

Let $M$ be a module over a commutative ring $R$. The annihilating-submodule graph of $M$, denoted by $AG(M)$, is a simple undirected 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 [H. Ansari-Toroghy and S. Habibi, 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 $M$.

Keywords

• Commutative ring
• annihilating-submodule graph
• domination number

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