A module theoretic approach to‎ ‎zero-divisor graph with respect to (first) dual

Abstract

Let $M$ be an $R$-module and $0 neq fin M^*={rm Hom}(M,R)$. We associate an undirected graph $gf$ to $M$ in which non-zero elements $x$ and $y$ of $M$ are adjacent provided that $xf(y)=0$ or $yf(x)=0$. We observe that over a commutative ring $R$, $gf$ is connected and diam$(gf)leq 3$. Moreover, if $Gamma (M)$ contains a cycle, then $mbox{gr}(gf)leq 4$. Furthermore if $|gf|geq 1$, then $gf$ is finite if and only if $M$ is finite. Also if $gf=emptyset$, then $f$ is monomorphism (the converse is true if $R$ is a domain). If $M$ is either a free module with ${rm rank}(M)geq 2$ or a non-finitely generated projective module there exists $fin M^*$ with ${rm rad}(gf)=1$ and ${rm diam}(gf)leq 2$. We prove that for a domain $R$ the chromatic number and the clique number of $gf$ are equal.