Abstract
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 \(S\) 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 \(S\)-acts. Furthermore, as an application of Nakayama’s lemma we prove that a projective \(S\)-act \(P\) is a projective cover for an \(S\)-act \(A\) if and only if \(P/P{\mathfrak {M}}\cong A/A{\mathfrak {M}},\) in which \({\mathfrak {M}}\) is the unique maximal right ideal of \(S\), it is two-sided and \(A\) is finitely generated quasi-strongly faithful.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ameri, R.: Two versions of Nakayama lemma for multiplication modules. Int. J. Math. Math. Sci. 54, 2911–2913 (2004)
Anderson, D.D., Johnson, E.W.: Ideal theory in commutative semigroups. Semigroup Forum 30, 127–158 (1984)
Anderson, D.D., Johnson, E.W.: Join-principally generated multiplicative lattices. Algebra Univers. 19, 74–82 (1984)
Anderson, F.W., Fuller, K.R.: Rings and Categories of Modules. Graduate Texts in Mathematics, 2nd edn. Springer, New York (1992)
Anjaneyulu, A.: Primary ideals in semigroups. Semigroup Forum 20, 129–144 (1980)
Azizi, A.: On generalization of Nakayama’s lemma. Glasg. Math. J. 52, 605–617 (2010)
Azumaya, G.: On maximally central algebras. Nagoya Math. J. 2, 119–150 (1951)
Balister, P., Howson, S.: Note on Nakayama’s lemma for compact \(\Lambda \)-Modules. Asian J. Math. 1, 224–229 (1997)
Cohen, P.M.: Further Algebra and Applications. Springer, New York (2003)
Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry. Graduate Texts in Mathemtaics. Springer, New York (1994)
Gilmer, R.: An exictence theorem for non-Noetherian rings. Am. Math. Mon. 77(6), 621–623 (1970)
Johnson, E.W.: Primary factorization in semigroups. Czechoslovak Math. J. 36(2), 180–184 (1986)
Kilp, M., Knauer, U., Mikhalev, A.V.: Monoids, Acts and Categories. Walter de Grupter, Berlin (2000)
Krull, W.: Dimensionstheorie in Stellenringen. J. Reine Angew. Math. 179, 204–226 (1938)
Nakayama, T.: A remark on finitely generated modules. Nagoya Math. J. 3, 139–140 (1951)
Ogus, A., Bergman, G.: Nakayama’s lemma for half-exact functors. Proc. Am. Math. Soc. 31, 67–74 (1972)
Sharp, R.Y.: Steps in Commutative Algebra, 2nd edn. Cambridge University Press, Cambridge (2000)
Zhang, X., Chen, Y., Wang, Y.: Chain conditions on essential subacts. Northeast Math. J. 22(3), 357–369 (2006)
Acknowledgments
The authors would like to thank Semnan university for its financial support.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Mohan Putcha.
Rights and permissions
About this article
Cite this article
Ahmadi, K., Madanshekaf, A. Nakayama’s lemma for acts over monoids. Semigroup Forum 91, 321–337 (2015). https://doi.org/10.1007/s00233-014-9653-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00233-014-9653-5