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

On (uniformly) $S$-$w$-Noetherian rings and modules