A note on the $S$-version of Noetherianity

TL;DR

This paper explores the properties of $S$-Noetherian rings and $S$-$ ext{ extasterisk}_w$-PIDs, providing negative answers to existing open questions and extending classical notions of Noetherianity and PID to these generalized contexts.

Contribution

It investigates $S$-Noetherian rings and $S$-$ ext{ extasterisk}_w$-PIDs, addressing open questions and broadening the understanding of these algebraic structures.

Findings

Negative answers to open questions on valuation domains.

Extension of classical Noetherian and PID concepts to $S$-versions.

Insights into the structure of $S$-Noetherian rings and related domains.

Abstract

It is well-known that a ring is Noetherian if and only if every ascending chain of ideals is stationary, and an integral domain is a PID if and only if every countably generated ideal is principal. We respectively investigate the similar results on $S$-Noetherian rings and $S$-$\ast_w$-PIDs, where $S$ is a multiplicative subset and $\ast$ is a star operation. In particular, we gave negative answers to the open questions proposed by Hamed and Hizem \cite{hh16}, Kim and Lim \cite{kl18}, and Lim \cite{l18} in terms of valuation domains, respectively.