Fiber decompositions and new invariants for zero-divisor graphs induced by unfaithful modules

Abstract

We study the zero-divisor graph [Formula: see text] attached to a commutative ring R and an unfaithful R-module M, with vertices the nonzero zero-divisors of R modulo I = [Formula: see text] and adjacency defined by annihilation on M. Starting from [Moh’d and Ahmed, Extending the Anderson–Livingston zero-divisor graph via unfaithful modules, Appl. Analysis Discrete Math, (2026)], we develop an exact fiber-decomposition theory that realizes [Formula: see text] as a mixed blow-up of [Formula: see text]. This viewpoint yields explicit formulas for the triangle number, clique number, independence number, chromatic number, diameter, girth, domination number, and several parity properties whenever I is finite. In particular, we characterize bipartiteness, regularity, and Eulerian behavior, and derive computable invariants from the quotient graph. Several concrete examples over [Formula: see text] and truncated polynomial rings illustrate the theory, while comparison tables and figures show how the annihilator ideal controls the passage from [Formula: see text] to [Formula: see text]. Our results suggest new problems on domination and planarity.