Zero-divisor graphs of nilpotent-free semigroups

TL;DR

This paper explores the relationships between zero-divisor graphs of nilpotent-free semigroups, rings, and topological spaces, introducing the Armendariz map to preserve graph invariants and reveal deep structural connections.

Contribution

It introduces the Armendariz map to relate zero-divisor graphs across various algebraic and topological structures, establishing new links and structure theorems.

Findings

Armendariz map preserves graph invariants across semigroups and rings

Relationships established between zero-divisor graphs and topological spectra

Strong structure theorems connecting algebraic, topological, and graph-theoretic properties

Abstract

We find strong relationships between the zero-divisor graphs of apparently disparate kinds of nilpotent-free semigroups by introducing the notion of an \emph{Armendariz map} between such semigroups, which preserves many graph-theoretic invariants. We use it to give relationships between the zero-divisor graph of a ring, a polynomial ring, and the annihilating-ideal graph. Then we give relationships between the zero-divisor graphs of certain topological spaces (so-called pearled spaces), prime spectra, maximal spectra, tensor-product semigroups, and the semigroup of ideals under addition, obtaining surprisingly strong structure theorems relating ring-theoretic and topological properties to graph-theoretic invariants of the corresponding graphs.