TY - JOUR
ID - 2730
TI - A KIND OF GRAPH STRUCTURE ASSOCIATED WITH ZERO-DIVISORS OF MONOID RINGS
JO - Journal of Algebraic Systems
JA - JAS
LA - en
SN - 2345-5128
AU - Etezadi, Mohammad
AU - Alhevaz, Abdollah
AD - Department of Pure Mathematics, Faculty of Mathematical Sciences, University of
Tabriz, Tabriz, Iran.
AD - Faculty of Mathematical Sciences, Shahrood University of Technology, P.O. Box
316-3619995161, Shahrood, Iran
Y1 - 2024
PY - 2024
VL - 11
IS - 2
SP - 53
EP - 63
KW - $M$-Armendariz graph
KW - diameter
KW - unique product monoid
KW - monoid ring
KW - domination number
DO - 10.22044/jas.2022.12238.1646
N2 - 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.
UR - https://jas.shahroodut.ac.ir/article_2730.html
L1 - https://jas.shahroodut.ac.ir/article_2730_aa5f7f98a22a1436fed3a41607d47007.pdf
ER -