Zero-Divisor Graphs of Quotient Rings

2015/08/10 by Rachael Alvir, Alvir, Rachael
Mathematics · #05C25 #Advanced Topics in Algebra #Commutative Algebra (math.AC) #FOS: Mathematics #Rings and Algebras (math.RA) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.1508.02432

openalex publication_date 2015/08/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The compressed zero-divisor graph Γ_C(R) associated with a commutative ring R has vertex set equal to the set of equivalence classes \ [r] | r ∈ Z(R), r ≠ 0 \ where r ∼ s whenever ann(r) = ann(s). Distinct classes [r],[s] are adjacent in Γ_C(R) if and only if xy = 0 for all x ∈ [r], y ∈ [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 Γ(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.

Related

• Exploring Ring Structures: Multiset Dimension Analysis in Compressed Zero-Divisor Graphs
• On the Zero-Divisor Graph of a Ring
• A Graph-Theoretical Approach to Ring Analysis: An Exploration of Dominant Metric Dimension in Compressed Zero Divisor Graphs and Its Interplay with Ring Structures
• The characteristic equation and Wiener index of a compressed zero divisor graph
• Induced subgraphs of zero-divisor graphs