Zero intersection graph of annihilator ideals of modules

Abstract

This paper aims to associate a new graph to nonzero unital modules over commutative rings. Let $R$ be a commutative ring having a nonzero identity and $M$ be a nonzero unital $R$-module. The zero intersection graph of annihilator ideals of $R$-module $M$, denoted by $\mathfrak{C}_{R}(M)$, is a simple (undirected) graph whose vertex set $M^{\star}=M-\{0\},\ $and two distinct vertices $m$ and $m^{\prime}$ are adjacent if $ann_{R}(m)\cap ann_{R}(m^{\prime})=(0).$\ We investigate the conditions under which $\mathfrak{C}_{R}(M)$ is a star graph, bipartite graph, complete graph, edgeless graph. Furthermore, we characterize certain classes of modules and rings such as torsion-free modules, torsion modules, semisimple modules, quasi-regular rings, and modules satisfying Property $T$ in terms of their graphical properties.

Keywords

• property A
• property T
• torsion-free module
• quasi regular ring
• weak quasi regular module

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