The triple zero graph of a commutative ring

Year 2021, Volume: 70 Issue: 2, 653 - 663, 31.12.2021

Ece Yetkin Çelikel

https://doi.org/10.31801/cfsuasmas.786804

Abstract

Let $R$ be a commutative ring with non-zero identity. We define the set of
triple zero elements of $R$ by $TZ(R)=\{a\in Z(R)^{\ast}:$ there exists
$b,c\in R\backslash\{0\}$ such that $abc=0$, $ab\neq0$, $ac\neq0$,
$bc\neq0\}.$ In this paper, we introduce and study some properties of the
triple zero graph of $R$ which is an undirected graph $TZ\Gamma(R)$ with
vertices $TZ(R),$ and two vertices $a$ and $b$ are adjacent if and only if
$ab\neq0$ and there exists a non-zero element $c$ of $R$ such that $ac\neq0$,
$bc\neq0$, and $abc=0$. We investigate some properties of the triple zero
graph of a general ZPI-ring $R,$ we prove that $diam(TZ\Gamma(R))\in\{0,1,2\}$
and $gr(G)\in\{3,\infty\}$.

Keywords

Triple zero graph, Zero-divisor graph, 2-absorbing ideal

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