GENERALIZATIONS OF THE ZERO-DIVISOR GRAPH

Authors : David F ANDERSON, Grace MCCLURKİN

Pages : 237-262

Doi:10.24330/ieja.663079

View : 13 | Download : 8

Publication Date : 2020-01-07

Article Type : Research Paper

Abstract :Let $R$ be a commutative ring with $1 \neq 0$ and $Zinsert ignore into journalissuearticles values(R);$ its set of zero-divisors. The zero-divisor graph of $R$ is the insert ignore into journalissuearticles values(simple); graph $\Gammainsert ignore into journalissuearticles values(R);$ with vertices $Zinsert ignore into journalissuearticles values(R); \setminus \{0\}$, and distinct vertices $x$ and $y$ are adjacent if and only if $xy = 0$. In this paper, we consider generalizations of $\Gammainsert ignore into journalissuearticles values(R);$ by modifying the vertices or adjacency relations of $\Gammainsert ignore into journalissuearticles values(R);$. In particular, we study the extended zero-divisor graph $\overline{\Gamma}insert ignore into journalissuearticles values(R);$, the annihilator graph $AGinsert ignore into journalissuearticles values(R);$, and their analogs for ideal-based and congruence-based graphs.
Keywords : Zero divisor graph, commutative ring with identity