On Z-Noetherian Rings
Abstract
In this paper, introduce a new finiteness condition for commutative rings, called Z-Noetherian, under which every ideal contained in the set of zero-divisors Z(R) is finitely generated. This condition lies properly between Noetherian and Nil*-Noetherian. We establish intrinsic characterizations of Z-Noetherian rings and investigate their stability under nilpotent quotients and localizations. Under the assumption that Z(R) is an ideal, we obtain a primary decomposition theorem for ideals contained in Z(R). For ZD-rings, we prove a Cohen-type criterion characterizing Z-Noetherian in terms of the finite generation of prime ideals contained in Z(R). We further establish Z-Noetherian analogues of the Hilbert Basis Theorem and Krull's Intersection Theorem. Finally, we establish a principal ideal theorem for Z-Noetherian rings. Collectively, these results show that several fundamental finiteness and structural properties of Noetherian rings extend to the broader class of Z-Noetherian rings.
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BibTeX
@misc{dubey2026znoetherian,
title = {{On Z-Noetherian Rings}},
author = {Ambar Dubey and Tushar Singh and Shiv Datt Kumar},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16178},
eprint = {2610.16178},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16178},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On Z-Noetherian Rings
AU - Ambar Dubey
AU - Tushar Singh
AU - Shiv Datt Kumar
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16178
UR - https://arxiv.org/abs/2610.16178
AB - In this paper, introduce a new finiteness condition for commutative rings, called Z-Noetherian, under which every ideal contained in the set of zero-divisors Z(R) is finitely generated. This condition lies properly between Noetherian and Nil_*-Noetherian. We establish intrinsic characterizations of Z-Noetherian rings and investigate their stability under nilpotent quotients and localizations. Under the assumption that Z(R) is an ideal, we obtain a primary decomposition theorem for ideals contained in Z(R). For ZD-rings, we prove a Cohen-type criterion characterizing Z-Noetherian in terms of the finite generation of prime ideals contained in Z(R). We further establish Z-Noetherian analogues of the Hilbert Basis Theorem and Krull's Intersection Theorem. Finally, we establish a principal ideal theorem for Z-Noetherian rings. Collectively, these results show that several fundamental finiteness and structural properties of Noetherian rings extend to the broader class of Z-Noetherian rings.
ER -