TY - JOUR ID - 13185 TI - On zero divisor graph of unique product monoid rings over Noetherian reversible ring JO - Categories and General Algebraic Structures with Applications JA - CGASA LA - en SN - 2345-5853 AU - Hashemi, Ebrahim AU - Alhevaz, Abdollah AU - Yoonesian, Eshag AD - Department of Mathematics, Shahrood University of Technology, Shahrood, Iran, P.O. Box: 316-3619995161. Y1 - 2016 PY - 2016 VL - 4 IS - 1 SP - 95 EP - 114 KW - Zero-divisor graphs KW - diameter KW - Girth KW - Reversible rings KW - Polynomial rings KW - Unique product monoids KW - Monoid rings DO - N2 -  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 $\Gamma(R[M])$. UR - https://cgasa.sbu.ac.ir/article_13185.html L1 - https://cgasa.sbu.ac.ir/article_13185_9afc1a95b9340cdc8d14a1cee3b2fe5c.pdf ER -