On the existence of parameterized noetherian rings

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.