Noetherian rings of non-local rank
Abstract
The rank of a ring is the supremum of minimal cardinalities of generating sets of , among all ideals in . In this paper, we obtain a characterization of Noetherian rings whose rank is not equal to the supremum of ranks of localizations of at maximal ideals. It turns out that any such ring is a direct product of a finite number of local principal Artinian rings and Dedekind domains, at least one of which is not a principal ideal ring. As an application, we show that the rank of the ring of polynomials over an Artinian ring can be computed locally.
Cite
@article{arxiv.2501.05940,
title = {Noetherian rings of non-local rank},
author = {Dmitry Kudryakov},
journal= {arXiv preprint arXiv:2501.05940},
year = {2025}
}
Comments
5 pages; provided a self-contained proof of the second inequality in Proposition 1.6 (now 1.8), expanded the proof of Lemma 2.2, added a footnote citing the arXiv preprint on the first page, removed "Noetherian" in Theorem 1.5, put Proposition 1.6 (with the proof and the remark before it) after Theorem 1.8, removed the first reference in the References section