When zero-divisor graphs are divisor graphs

Authors : Emad Abu OSBA, Osama ALKAM

Pages : 797-807

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Publication Date : 0000-00-00

Article Type : Research Paper

Abstract :Let $R$ be a finite commutative principal ideal ring with unity. In this article, we prove that the zero-divisor graph $\Gammainsert ignore into journalissuearticles values(R);$ is a divisor graph if and only if $R$ is a local ring or it is a product of two local rings with at least one of them having diameter less than $2$. We also prove that $\Gammainsert ignore into journalissuearticles values(R);$ is a divisor graph if and only if $\Gammainsert ignore into journalissuearticles values(R[x]);$ is a divisor graph if and only if $\Gammainsert ignore into journalissuearticles values(R[[x]]);$ is a divisor graph.
Keywords : Principal ideal ring, zero divisor graph, divisor graph, polynomial ring, power series ring