A Note on S-Nakayama’s Lemma

Abstract

We propose an $S$-version of Nakayama's lemma. Let $R$ be a commutative ring, $S$ a multiplicative subset of $R,$ and $M$ be an $S$-finite $R$-module. Also let $I$ be an ideal of $R.$ We show that if there exists $t\in S$ such that $tM\subseteq IM,$ then$(t^\prime+a)M=0$ for some $t'\in S$ and $a\in I.$ We also give an analog of Nakayama's lemma for a $w$-ideal and an $S$-$w$-finite $R$-module, where $R$ is an integral domain. Thus, we generalize the result obtained by Wang and McCasland [Commun. Algebra, 25, 1285–1306 (1997)].