On the existence of parameterized noetherian rings

2025/04/14 by Zhang, Xiaolei
#FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2504.09822

Abstract

A ring R is called left strictly (<ℵ_α)-noetherian if ℵ_α is the minimum cardinal such that every ideal of R is (<ℵ_α)-generated. In this note, we show that for every singular (resp., regular) cardinal ℵ_α, there is a valuation domain D, which is strictly (<ℵ_α)-noetherian (resp., strictly (<ℵ_α⁺)-noetherian), positively answering a problem proposed in \citeMarcos25 under some set theory assumption.

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