Journal of the
Korean Mathematical Society JKMS

Let R be a commutative ring with identity and $M$ an $R$-module. In this paper, we associate a graph to $M$, say $\g$, such that when $M=R$, $\g$ is exactly the classic zero-divisor graph. Many well-known results by D. F. Anderson and P. S. Livingston, in \cite{AL}, and by D. F. Anderson and S. B. Mulay, in \cite{AM}, have been generalized for $\g$ in the present article. We show that $\g$ is connected with ${\rm{diam}}(\g)\leq 3$. We also show that for a reduced module $M$ with $\z \neq M\setminus \{0\}$, ${\rm{gr}}(\g)=\infty$ if and only if $\g$ is a star graph. Furthermore, we show that for a finitely generated semisimple $R$-module $M$ such that its homogeneous components are simple, $x, y\in M\setminus\{0\}$ are adjacent if and only if $xR\bigcap yR=(0)$. Among other things, it is also observed that $\g=\emptyset$ if and only if $M$ is uniform, ${\rm{ann}}(M)$ is a radical ideal, and $\z\neq M\setminus \{0\}$, if and only if ${\rm{ann}}(M)$ is prime and $\z\neq M\setminus \{0\}$.

Keywords: module, zero-divisor graph of modules, girth, diameter, complete bipartite graph

MSC numbers: 05C25, 05C38, 05C40 16D10, 16D4