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

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 α=a1g1+⋯‎+‎angn and β=b1h1+⋯+bmhm are adjacent if and only if aibj=0 or bjai=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 Γ(R[M])‎, ‎where R is a reversible ring and M is a unique product monoid‎.

Keywords

References

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Journal of Algebraic Systems
Volume 11, Issue 2
January 2024
Pages 53-63

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