Hashemi, E., Alhevaz, A., Yoonesian, E. (2016). On zero divisor graph of unique product monoid rings over Noetherian reversible ring. Categories and General Algebraic Structures with Applications, 4(1), 95-114.

Ebrahim Hashemi; Abdollah Alhevaz; Eshag Yoonesian. "On zero divisor graph of unique product monoid rings over Noetherian reversible ring". Categories and General Algebraic Structures with Applications, 4, 1, 2016, 95-114.

Hashemi, E., Alhevaz, A., Yoonesian, E. (2016). 'On zero divisor graph of unique product monoid rings over Noetherian reversible ring', Categories and General Algebraic Structures with Applications, 4(1), pp. 95-114.

Hashemi, E., Alhevaz, A., Yoonesian, E. On zero divisor graph of unique product monoid rings over Noetherian reversible ring. Categories and General Algebraic Structures with Applications, 2016; 4(1): 95-114.

On zero divisor graph of unique product monoid rings over Noetherian reversible ring

^{}Department of Mathematics, Shahrood University of Technology, Shahrood, Iran, P.O. Box: 316-3619995161.

Abstract

Let $R$ be an associative ring with identity and $Z^*(R)$ be its set of non-zero zero divisors. The zero-divisor graph of $R$, denoted by $\Gamma(R)$, is the graph whose vertices are the non-zero zero-divisors of $R$, and two distinct vertices $r$ and $s$ are adjacent if and only if $rs=0$ or $sr=0$. In this paper, we bring some results about undirected zero-divisor graph of a monoid ring over reversible right (or left) Noetherian ring $R$. We essentially classify the diameter-structure of this graph and show that $0\leq \mbox{diam}(\Gamma(R))\leq \mbox{diam}(\Gamma(R[M]))\leq 3$. Moreover, we give a characterization for the possible diam$(\Gamma(R))$ and diam$(\Gamma(R[M]))$, when $R$ is a reversible Noetherian ring and $M$ is a u.p.-monoid. Also, we study relations between the girth of $\Gamma(R)$ and that of $\Gamma(R[M])$.

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