Multiplicity-One Structure in Graded Zero-Divisor Graphs
Abstract
Let H be a multiplicative group, R a commutative H-graded ring, and M a nonzero finitely generated gr-semisimple H-graded R-module. We assume that M admits a fixed decomposition into powers of gr-simple modules with pairwise distinct annihilators, each repeated block carrying the componentwise grading. For a nonzero homogeneous element x, we compute the colon ideal (Rx:M) and show that it is determined by the multiplicity-one blocks on which x has a nonzero component. This leads to the criterion x↔ y if and only if σ(x)∩σ(y)=∅, where x↔ y means that x and y are adjacent in ΓH(M). Thus adjacency is governed by disjointness of multiplicity-one obstruction supports. The corresponding obstruction classes form a weighted disjointness structure: higher multiplicities affect the sizes of the classes, but not the adjacency relation between their supports. We use this description to study graph invariants, reconstruction, and Kneser-type layers, and, in the finite case, to obtain formulas for the obstruction-class sizes and a reduction of the adjacency spectrum.
Related
- Zero-divisor graphs and zero-divisor real rank of C*-algebras
- Metric Bases of Barycentric and Matching Subdivisions of Zero-Divisor Graphs
- Havel--Hakimi Residues of Common-Divisor Graphs: Complete Asymptotics and Prime-Counting Structure
- Asymptotic multiplicity sequences of graded families
- The zero divisor and augmentation conjectures for medial quandle rings
BibTeX
@misc{aijaz2026multiplicityone,
title = {{Multiplicity-One Structure in Graded Zero-Divisor Graphs}},
author = {M. Aijaz and Shahnawaz A. Rather and S Pirzada},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14003},
eprint = {2610.14003},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14003},
}
RIS
TY - UNPB
TI - Multiplicity-One Structure in Graded Zero-Divisor Graphs
AU - M. Aijaz
AU - Shahnawaz A. Rather
AU - S Pirzada
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.14003
UR - https://arxiv.org/abs/2610.14003
AB - Let H be a multiplicative group, R a commutative H-graded ring, and M a nonzero finitely generated gr-semisimple H-graded R-module. We assume that M admits a fixed decomposition into powers of gr-simple modules with pairwise distinct annihilators, each repeated block carrying the componentwise grading. For a nonzero homogeneous element x, we compute the colon ideal (Rx:M) and show that it is determined by the multiplicity-one blocks on which x has a nonzero component. This leads to the criterion x↔ y if and only if σ(x)∩σ(y)=∅, where x↔ y means that x and y are adjacent in Γ_H(M). Thus adjacency is governed by disjointness of multiplicity-one obstruction supports. The corresponding obstruction classes form a weighted disjointness structure: higher multiplicities affect the sizes of the classes, but not the adjacency relation between their supports. We use this description to study graph invariants, reconstruction, and Kneser-type layers, and, in the finite case, to obtain formulas for the obstruction-class sizes and a reduction of the adjacency spectrum.
ER -