  Let $R$ be a ring and $S$ a multiplicative subset of $R$. We introduce and
study the notions of ($u$-)$S$-$w$-Noetherian modules and
($u$-)$S$-$w$-principal ideal modules. Some characterizations of these new
concepts are given. In particular, we positively answer two open questions
proposed by Kim and Lim [12, Question 14] and Lim [8, Question 8].
