Zero-Divisor Graphs of Quotient Rings

TL;DR

This paper studies the structure of compressed zero-divisor graphs of quotient rings of UFDs, providing methods to construct these graphs and conditions for their isomorphism, with conjectures on their complete characterization.

Contribution

It introduces a construction method for compressed zero-divisor graphs of quotient rings of UFDs and establishes conditions for graph isomorphism, including conjectures for a full characterization.

Findings

Methods to construct zero-divisor graphs for principal ideal quotients.

Sufficient conditions for graph isomorphism of quotient rings.

Conjectures on necessary and sufficient conditions for isomorphism with loops.

Abstract

The compressed zero-divisor graph $\Gamma_C(R)$ associated with a commutative ring $R$ has vertex set equal to the set of equivalence classes $\{ [r] \mid r \in Z(R), r \neq 0 \}$ where $r \sim s$ whenever $ann(r) = ann(s)$. Distinct classes $[r],[s]$ are adjacent in $\Gamma_C(R)$ if and only if $xy = 0$ for all $x \in [r], y \in [s]$. In this paper, we explore the compressed zero-divisor graph associated with quotient rings of unique factorization domains. Specifically, we prove several theorems which exhibit a method of constructing $\Gamma(R)$ for when one quotients out by a principal ideal, and prove sufficient conditions for when two such compressed graphs are graph-isomorphic. We show these conditions are not necessary unless one alters the definition of the compressed graph to admit looped vertices, and conjecture necessary and sufficient conditions for two compressed graphs with loops to be isomorphic when considering any quotient ring of a unique factorization domain.