Domination Number in the Annihilating-Submodule Graph of Modules Over Commutative Rings

Abstract

Let M be a module over a commutative ring R. The annihilatingsubmodule 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 nonzero proper submodule K of M such that N K = (0), where N K, 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 N K = (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(τ 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 .

Some Developments in the Field of Homological Algebra by Defining New Class of Modules over Nonassociative Rings

“…Some further developments in the field of modules were done by Ansari and Habib in [24] by defining their graphs over rings.…”