On the graph of modules over commutative rings

Abstract

Let $M$ be a module over a commutative ring and let $\Spec (M)$ be the collection of all prime submodules of $M$. We topologize $\Spec (M)$ with quasi-Zariski topology and, for a subset $T$ of $\Spec (M)$, we introduce a new graph $G(\tau ^{*}_{T})$, called the \textit {quasi-Zariski topology-graph}. It helps us to study algebraic (respectively, topological) properties of $M$ (respectively, $\Spec (M)$) by using graph theoretical tools. Also, we study the annihilating-submodule graph and investigate the relation between these two graphs.