Zero-Divisor Graphs and Zero-Divisor Functors
Enrico Sbarra, Maurizio Zanardo

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.
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Taxonomy
TopicsRings, Modules, and Algebras · Homotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models
