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

Document Type: Research Paper

Authors

Department of Mathematics‎, ‎Yasouj University‎, ‎Yasouj,75914‎, ‎Iran.

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.

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