In this paper we present a new graph that is closely related to the classical zero-divisor graph. In our case two nonzero distinct zero divisors x and y of a commutative ring R are adjacent whenever there exist two nonnegative integers n and m such that x n y m = 0 with x n ̸ = 0 and y m ̸ = 0. This yields an extension of the classical zero divisor graph Γ(R) of R , which will be denoted by Γ(R). First we distinguish when Γ(R) and Γ(R) coincide. Various examples in this context are given. We show that if Γ(R) ̸ = Γ(R) , then Γ(R) must contain a cycle. We also show that if Γ(R) ̸ = Γ(R) and Γ(R) is complemented, then the total quotient ring of R is zero-dimensional. Among other things, the diameter and girth of Γ(R) are also studied.
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.
The extended zero-divisor graph and the annihilator graph of a ring are two extensions of the classical zero-divisor graph. In this paper, we investigate the relation between these graphs. Relation between these graphs on some particular ring constructions is also given.
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