Sep 2026· Journal of Algebra and its Applications· 0 citations
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.
Let R be a finite commutative ring with identity, and let Γ′ᵣ(R) denote its reduced cozero-divisor graph. We organize Γ′ᵣ(R) through the poset Prin*(R) of nonzero proper principal ideals and prove that adjacency is exactly incomparability in this poset. Consequently, cliques correspond to antichains, independent sets c...
Dong-Ze Du· Theoretical and Natural Scie...· 0 citations
Let R be a finite commutative ring with identity and let I be a proper ideal of R. The ideal-based zero-divisor graph \Gamma_I(R) has vertices outside I that annihilate some element outside I modulo I, with x adjacent to y whenever xy\in I. This paper studies the adjacency energy of \Gamma_I(R). General trace bounds ar...
Rosalio G. Artes, R. Malalay, M. Mbah et al.· International Journal of Mat...· 0 citations
The zero divisor graph $\Gamma(R)$ of a finite commutative ring $R$ has as vertices the non-zero zero divisors of $R$, with an edge between two elements exactly when their product is zero. We determine the boxicity and threshold dimension of $\Gamma(R)$ for two classes of finite commutative rings: reduced rings and quo...
The zero-divisor graph of a commutative ring provides a natural connection between algebraic and graph-theoretic structures. Although extensive research has been conducted on the algebraic and combinatorial properties of zero-divisor graphs, their connectivity and metric properties over finite semilocal rings remain co...
Presley Kiplagat· Earthline Journal of Mathema...· 0 citations
Yang’s linear tridle construction assigns a two-variable module to an oriented knot diagram. We show that the entire module, not only its maximal-minor polynomial, is Alexander-theoretic. A signed change of regional generators using the Alexander numbering transforms the linear tridle matrix into the Alexander–Dehn reg...
Ming-Hui Liu, Bo-Xin Zhou· Journal of knot theory and i...· 0 citations
This article examines the homological invariants, including Castelnuovo-Mumford regularity, projective dimension, and Betti numbers, of the edge ideals associated with the power graphs of integer modulo groups. We characterize the edge ideals of power graphs of group
\mathbb{Z}_{n}
with 2-linear resolution and list...
B. Rather, Jian-Feng Wang· Filomat· 2 citations· ⚡1
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
