非局部秩的Noetherian环
交换代数
2025-09-22 v3
摘要
环的秩是所有理想的生成集的最小基数中的上确界。本文刻画了秩不等于其极大理想处局部化秩的上确界的Noetherian环。结果表明，任何这样的环都是有限个局部主Artin环和Dedekind域的直接乘积，其中至少有一个不是主理想环。作为应用，我们证明了Artin环上多项式环的秩可以局部计算。
引用
@article{arxiv.2501.05940,
title = {Noetherian rings of non-local rank},
author = {Dmitry Kudryakov},
journal= {arXiv preprint arXiv:2501.05940},
year = {2025}
}
备注
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