A crucial lemma on module theory is Nakayama’s lemma (Anderson and Fuller in Rings and Categories of Modules, Springer, New York, 1992). In this manuscript, we shall investigate some forms of Nakayama’s lemma in the category of right acts over a given monoid S\documentclass[12pt]{minimal}
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\begin{document}$$S$$\end{document} with identity 1. In fact we present two forms of which the latter is similar to that on modules. To this end, we introduce the notion of quasi-strongly faithful for acts which is more general than that of strongly faithful which exists in the context. Some relevant examples are indicated. Among other things, we prove Krull intersection theorem for S\documentclass[12pt]{minimal}
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\begin{document}$$S$$\end{document}-acts. Furthermore, as an application of Nakayama’s lemma we prove that a projective S\documentclass[12pt]{minimal}
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\begin{document}$$S$$\end{document}-act P\documentclass[12pt]{minimal}
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\begin{document}$$P$$\end{document} is a projective cover for an S\documentclass[12pt]{minimal}
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\begin{document}$$S$$\end{document}-act A\documentclass[12pt]{minimal}
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\begin{document}$$A$$\end{document} if and only if P/PM≅A/AM,\documentclass[12pt]{minimal}
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\begin{document}$$P/P{\mathfrak {M}}\cong A/A{\mathfrak {M}},$$\end{document} in which M\documentclass[12pt]{minimal}
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\begin{document}$${\mathfrak {M}}$$\end{document} is the unique maximal right ideal of S\documentclass[12pt]{minimal}
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\begin{document}$$S$$\end{document}, it is two-sided and A\documentclass[12pt]{minimal}
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\begin{document}$$A$$\end{document} is finitely generated quasi-strongly faithful.
