On the existence of parameterized noetherian rings

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.