The triple zero graph of a commutative ring

Authors : Ece YETKİN ÇELİKEL

Pages : 653-663

Doi:10.31801/cfsuasmas.786804

View : 15 | Download : 18

Publication Date : 2021-12-31

Article Type : Research Paper

Abstract :Let $R$ be a commutative ring with non-zero identity. We define the set of triple zero elements of $R$ by $TZinsert ignore into journalissuearticles values(R);=\{a\in Zinsert ignore into journalissuearticles values(R);^{\ast}:$ 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 $TZ\Gammainsert ignore into journalissuearticles values(R);$ with vertices $TZinsert ignore into journalissuearticles values(R);,$ 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 $diaminsert ignore into journalissuearticles values(TZ\Gammainsert ignore into journalissuearticles values(R););\in\{0,1,2\}$ and $grinsert ignore into journalissuearticles values(G);\in\{3,\infty\}$.
Keywords : Triple zero graph, Zero divisor graph, 2 absorbing ideal