Zero-Divisor Graphs and Zero-Divisor Functors

TL;DR

This paper introduces a generalized framework for zero-divisor graphs of rings using functors, unifying and extending existing definitions and results in the literature.

Contribution

It defines a new class of zero-divisor graphs via functors, providing a unifying and general framework that encompasses all classical definitions.

Findings

Unified zero-divisor graph framework

Generalization of classical results

Potential for further theoretical developments

Abstract

Inspired by a very recent work of A. {\DH}uri\'c, S. Jev{\dj}eni\'c and N. Stopar, we introduce a new definition of zero-divisor graphs attached to rings, that includes all of the classical definitions already known in the literature. We provide an interpretation of such graphs as images of a functor, that we call zero-divisor functor and which is associated with a family of special equivalence relations fixed beforehand. We thus recover and generalize many known results for zero-divisor graphs and provide a framework which might be useful for further investigations on this topic.