Document Type : Original Manuscript
Authors
1 Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran.
2 Faculty of Mathematical Sciences, Shahrood University of Technology, P.O. Box 316-3619995161, Shahrood, Iran
Abstract
Let $R$ be an associative ring and $M$ be a monoid. In this paper, we introduce new kind of graph structure asociated with zero-divisors of monoid ring $R[M]$, calling it the $M$-Armendariz graph of a ring $R$ and denoted by $A(R,M)$. It is an undirected graph whose vertices are all non-zero zero-divisors of the monoid ring $R[M]$ and two distinct vertices $\alpha=a_{1}g_{1}+\cdots+ a_{n}g_{n}$ and $\beta=b_{1}h_{1}+\cdots+b_{m}h_{m}$ are adjacent if and only if $a_{i}b_{j}=0$ or $b_{j}a_{i}=0$ for all $i,j$. We investigate some graph properties of $A(R,M)$ such as diameter, girth, domination number and planarity. Also, we get some relations between diameters of the $M$-Armendariz graph $A(R,M)$ and that of zero divisor graph $\Gamma(R[M])$, where $R$ is a reversible ring and $M$ is a unique product monoid.
Keywords
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