Quasi-projective covers of right $S$-acts

Document Type : Research Paper


Department of Mathematics, College of Science, Shiraz University, Shiraz 71454, Iran.


In this paper $S$ is a monoid with a left zero and $A_S$ (or $A$) is a unitary right $S$-act. It is shown that a monoid $S$ is right perfect (semiperfect) if and only if every (finitely generated) strongly flat right $S$-act is quasi-projective. Also it is shown that if every right $S$-act has a unique zero element, then the existence of a quasi-projective cover for each right act implies that every right act has a projective cover. ‎  


[1] J. Ahsan and K. Saifullah, Completely quasi-projective monoids, Semigroup Forum 38 (1989), 123-126.
[2] J. Fountain, Perfect semigroups, Proc. Edinburgh Math. Soc. 20(3) (1976), 87-93.
[3] J. Isbell, Perfect monoids, Semigroup Forum 2 (1971), 95-118.
[4] R. Khosravi, M. Ershad, and M. Sedaghatjoo, Storngly  at and condition (P) covers of acts over monoids, Comm. Algebra 38(12) (2010), 4520-4530.
[5] M. Kilp, U. Knauer, and A. Mikhalev, \Monoids, Acts and Categories, With Application to Wreath Products and Graphs"; Berlin, New York, 2000.
[6] U. Knauer and H. Oltmanns, On Rees weakly projective right acts, J. Math. Sci. 139(4) (2006), 6715-6722.
[7] U. Knauer and H. Oltmanns, Weak projectivities for S-acts, Proceeding of the Conference on General Algebra and Discrete Math. (postsdam), Aachen (1999), 143-159.
[8] M. Mahmoudi and J. Renshaw, On covers of cyclic acts over monoids, Semigroup Forum 77 (2008), 325-338.
[9] J. Wei, On a question of Kilp and Knauer, Comm. Algebra 32(6) (2004), 2269-2272.