A note on the $S$-version of Noetherianity

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.