Journal of the
Korean Mathematical Society JKMS

Let $M$ be an $R$-module, where $R$ is a commutative ring with identity $1$ and let $G(V,E)$ be a graph. In this paper, we study the graphs associated with modules over commutative rings. We associate three simple graphs $ann_f(\Gamma(M_R))$, $ann_s(\Gamma(M_R))$ and $ann_t(\Gamma(M_R))$ to $M$ called full annihilating, semi-annihilating and star-annihilating graph. When $M$ is finite over $R$, we investigate metric dimensions in $ann_f(\Gamma(M_R))$, $ann_s(\Gamma(M_R))$ and $ann_t(\Gamma(M_R))$. We show that $M$ over $R$ is finite if and only if the metric dimension of the graph $ann_f(\Gamma(M_R))$ is finite. We further show that the graphs $ann_f(\Gamma(M_R))$, $ann_s(\Gamma(M_R))$ and $ann_t(\Gamma(M_R))$ are empty if and only if $M$ is a prime-multiplication-like $R$-module. We investigate the case when $M$ is a free $R$-module, where $R$ is an integral domain and show that the graphs $ann_f(\Gamma(M_R))$, $ann_s(\Gamma(M_R))$ and $ann_t(\Gamma(M_R))$ are empty if and only if $M \cong R$. Finally, we characterize all the non-simple weakly virtually divisible modules $M$ for which $Ann(M)$ is a prime ideal and $Soc(M) = 0$.

Keywords: module, zero-divisor graph, ring, metric dimension, multi\-plication-like module

MSC numbers: Primary 13A99, 05C99, 13C99