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A crucial lemma on module theory is Nakayama's lemma \cite{AF}. In thisarticle, we shall investigate some forms of Nakayama's lemma in the category ofright acts over a given monoid $S$ with identity 1. More precisely, among otherthings, 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 itsunique maximal right ideal $\mathfrak{M}$ is two-sided. Furthermore, as anapplication of Nakayama's lemma we prove Krull intersection theorem for$S$-acts. Finally, as a consequence, we shall see a homological classificationform of this lemma, i.e, we prove if $S$ is a commutative monoid then everyprojective $S$-act is free if and only if $E(S)=\{1\}$, which $E(S)$ is the setof all idempotents of $S$.

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