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 $M$.
Highlights
Throughout this paper R is a commutative ring with a non-zero identity and M is a unital R-module
We prove that the domination number of AG(M ) is equal to the number of factors in the Artinian decomposition of M and we find several domination parameters of AG(M )
We study the domination number of the annihilating-submodule graphs for reduced rings with finitely many minimal primes and faithful modules
Summary
Throughout this paper R is a commutative ring with a non-zero identity and M is a unital R-module. 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 N [U ] = N (U ) ∪ {U }. Let S be a dominating set of a graph G, and u ∈ S.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
