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This is a Quantitative Study study.

September 30, 2025Open Access

On study of algebraic structures: multiset dimensions in zero-divisor graphs associated with rings

Key Points

Findings establish general bounds for multiset dimension in zero divisor graphs, enhancing algebraic understanding.
A complete characterization of multiset dimension for Z_n is provided, emphasizing structural patterns in rings.
The study reveals important connections between algebraic structures and isomorphism through multiset dimensions.
The results extend to various rings, including Gaussian integers and quotient polynomial rings, highlighting their significance.

Abstract

This article explores the multiset dimension (Mdim) in zero divisor graphs (ZD-graphs) of commutative rings R with unity. Given a finite ring, its zero divisors form the set L (R), which defines the ZD-graph. We establish general bounds for Mdim in ZD-graphs and extend these results to various rings, including Gaussian integers, Ring of Zₙ modulo n and quotient polynomial rings. Notably, we provide a complete characterization of Mdim for Zₙ for all n. Our findings reveal structural patterns among rings with identical Mdim, emphasizing its role in isomorphism and enhancing the algebraic understanding of these graphs.

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