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

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 Z_n modulo n and quotient polynomial rings. Notably, we provide a complete characterization of Mdim for Z_n ​ 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.