On the existence of parameterized noetherian rings
Xiaolei Zhang
TL;DR
This paper addresses the existence of strictly $(<\aleph_\alpha)$-noetherian rings for limit cardinals, a question posed by Marcos25. It constructs a valuation domain $D$ with valuation group $G=\bigoplus_{i\in\aleph_\alpha}\mathbb{Q}$ and analyzes the generator cardinalities of its ideals. The main results separate the regular and singular cases: when $\aleph_\alpha$ is regular, $D$ is strictly $(<\aleph_\alpha^{+})$-noetherian; when $\aleph_\alpha$ is singular, $D$ is strictly $(<\aleph_\alpha)$-noetherian. Under the set-theoretic hypothesis that weakly inaccessible cardinals do not exist, these constructions provide a positive answer to Marcos25's Problem 2.6, illustrating the dependence on large-cardinal assumptions in these algebraic questions.
Abstract
A ring $R$ is called left strictly $(<\aleph_α)$-noetherian if $\aleph_α$ is the minimum cardinal such that every ideal of $R$ is $(<\aleph_α)$-generated. In this note, we show that for every singular (resp., regular) cardinal $\aleph_α$, there is a valuation domain $D$, which is strictly $(<\aleph_α)$-noetherian (resp., strictly $(<\aleph_α^+)$-noetherian), positively answering a problem proposed in \cite{Marcos25} under some set theory assumption.