The extended zero-divisor graph of the amalgamated duplication of a ring along an ideal

TL;DR

This paper explores the properties of the extended zero-divisor graph of the amalgamated duplication of a ring along an ideal, focusing on graph coincidence, completeness, diameter, and girth.

Contribution

It characterizes when the extended and standard zero-divisor graphs coincide, when the graph is complete, and computes key graph invariants for the amalgamated duplication.

Findings

Identifies conditions for graph coincidence

Provides criteria for graph completeness

Calculates diameter and girth of the extended zero-divisor graph

Abstract

Let $R$ be a commutative ring and $I$ be an ideal of $R$. The amalgamated duplication of $R$ along $I$ is the subring $R\Join I:=\{(r,r+i)| r\in R, i\in I\}$ of $R\times R$. This paper investigates the extended zero-divisor graph of the amalgamated duplication of $R$ along $I$. The purpose of this work is to study when $\overline{\Gamma}(R\Join I)$ and $\Gamma(R\Join I)$ coincide, to characterize when $\overline{\Gamma}(R\Join I)$ is complete, and to compute the diameter and the girth of $\overline{\Gamma}(R\Join I)$.