GENERALIZATIONS OF THE ZERO-DIVISOR GRAPH

Year 2020, , 237 - 262, 07.01.2020

David F. Anderson Grace Mcclurkin

https://doi.org/10.24330/ieja.663079

Cited By: 5

Abstract

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

Keywords

Zero-divisor graph, commutative ring with identity

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