On graphs associated with modules over commutative rings
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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 $Soc(M) = 0$.
Abstract
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 $Soc(M) = 0$.
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