Nakayama's Lemma on $\textbf{Act}-S$

Abstract

A crucial lemma on module theory is Nakayama's lemma \cite{AF}. In this article, we shall investigate some forms of Nakayama's lemma in the category of right acts over a given monoid $S$ with identity 1. More precisely, among other things, we show that equality $AI=A$ for some proper ideal $I$ of $S$ implies $A=\{\theta\}$, when $A$ is a finitely generated quasi-strongly faithful $S$-act with unique zero element $\theta$ and $S$ is a monoid in which its unique maximal right ideal $\mathfrak{M}$ is two-sided. Furthermore, as an application of Nakayama's lemma we prove Krull intersection theorem for $S$-acts. Finally, as a consequence, we shall see a homological classification form of this lemma, i.e, we prove if $S$ is a commutative monoid then every projective $S$-act is free if and only if $E(S)=\{1\}$, which $E(S)$ is the set of all idempotents of $S$.