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

Authors : Habibollah ANSARITOROGHY, Shokoufeh HABIBI

Pages : 203-216

Doi:10.24330/ieja.969902

View : 19 | Download : 11

Publication Date : 2021-07-17

Article Type : Research Paper

Abstract :Let $M$ be a module over a commutative ring $R$. The annihilating-submodule graph of $M$, denoted by $AGinsert ignore into journalissuearticles values(M);$, 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 $NK=insert ignore into journalissuearticles values(0);$, where $NK$, the product of $N$ and $K$, is denoted by $insert ignore into journalissuearticles values(N:M);insert ignore into journalissuearticles values(K:M);M$ and two distinct vertices $N$ and $K$ are adjacent if and only if $NK=insert ignore into journalissuearticles values(0);$. 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 $Ginsert ignore into journalissuearticles values(\tau_T);$ which was introduced in [H. Ansari-Toroghy and S. Habibi, Comm. Algebra, 42insert ignore into journalissuearticles values(2014);, 3283-3296]. In this paper, we study the domination number of $AGinsert ignore into journalissuearticles values(M);$ and some connections between the graph-theoretic properties of $AGinsert ignore into journalissuearticles values(M);$ and algebraic properties of module $M$.
Keywords : Commutative ring, annihilating submodule graph, domination number