Extended Zero-Divisor Graphs of Idealizations

Abstract

Abstract. Let R be a commutative ring with zero-divisors Z(R). The extended zero-divisor graph of R, denoted by Γ(R), is the (simple) graph with vertices Z(R) * = Z(R)\{0}, the set of nonzero zero-divisors of R, where two distinct nonzero zero-divisors x and y are adjacent whenever there exist two non-negative integers n and m such that x n y m = 0 with x n = 0 and y m = 0. In this paper, we consider the extended zero-divisor graphs of idealizations R ⋉ M (where M is an R-module). At first, we distinguish when Γ(R ⋉ M ) and the classical zero-divisor graph Γ(R ⋉ M ) coincide. Various examples in this context are given. Among other things, the diameter and the girth of Γ(R ⋉ M ) are also studied.

Induced subgraphs of zero-divisor graphs

“…The zero-divisor graph Γ(R) of a ring R has vertices the zero-divisors in R (the non-zero elements a for which there exists b = 0 such that ab = 0), with an edge {a, b} whenever ab = 0. The zero-divisor graphs have been extensively studied in the past [1,4,5,6,7,8,10,11,19,21,23].…”

A note on zero-divisor graph of amalgamated duplication of a ring along an ideal

“…For a better understanding of the concept of zero-divisor graphs, several authors have been interested in studying zero-divisor graphs of certain ring constructions (see for example [2,3,6,9]). In this paper, we are interested in studying the extended zero-divisor graph of the amalgamated duplication of a ring along an ideal.…”

On two extensions of the classical zero-divisor graph

“…and moreover it is a complete graph. Also one can consider the ring Z Z p for any prime positive integer p (see [17]). 2.…”