مرکز اطلاعات علمی Scientific Information Database (SID) - Trusted Source for Research and Academic Resources

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Information Journal Paper

Title

A KIND OF GRAPH STRUCTURE ASSOCIATED WITH ZERO-DIVISORS OF MONOID RINGS

Pages

  53-63

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 .

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