The zero-divisor graph of an amalgamated algebra

TL;DR

This paper studies the zero-divisor graph of amalgamated algebras, providing new characterizations of its properties and calculating the diameter in specific cases, thereby advancing understanding of algebraic graph structures.

Contribution

It generalizes recent results on zero-divisor graphs of amalgamated algebras and offers new characterizations and diameter calculations, especially for finite rings.

Findings

Characterization of completeness of zero-divisor graphs

Complete description of diameter for finite rings

Improved bounds and formulas for the diameter

Abstract

Let $R$ and $S$ be commutative rings with identity, $f:R\to S$ a ring homomorphism and $J$ an ideal of $S$. Then the subring $R\bowtie^fJ:=\{(r,f(r)+j)\mid r\in R$ and $j\in J\}$ of $R\times S$ is called the amalgamation of $R$ with $S$ along $J$ with respect to $f$. In this paper, we generalize and improve recent results on the computation of the diameter of the zero-divisor graph of amalgamated algebras and obtain new results. In particular, we provide new characterizations for completeness of the zero-divisor graph of amalgamated algebra, as well as, a complete description for the diameter of the zero-divisor graph of amalgamations in the special case of finite rings.