Zero-Divisor Graphs of Quotient Rings
Rachael Alvir

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 associated with a commutative ring has vertex set equal to the set of equivalence classes where whenever . Distinct classes are adjacent in if and only if for all . 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 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…
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Taxonomy
TopicsRings, Modules, and Algebras · Advanced Topics in Algebra
