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arXiv · 2504.09822

On the existence of parameterized noetherian rings

Xiaolei Zhang
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Abstract

A ring $R$ is called left strictly $(<\aleph_{\alpha})$-noetherian if $\aleph_{\alpha}$ is the minimum cardinal such that every ideal of $R$ is $(<\aleph_{\alpha})$-generated. In this note, we show that for every singular (resp., regular) cardinal $\aleph_{\alpha}$, there is a valuation domain $D$, which is strictly $(<\aleph_{\alpha})$-noetherian (resp., strictly $(<\aleph_{\alpha}^+)$-noetherian), positively answering a problem proposed in \cite{Marcos25} under some set theory assumption.

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Xiaolei Zhang. 2025-04-14. On the existence of parameterized noetherian rings. https://arxiv.org/abs/2504.09822

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