$M$-projective and strongly $M$-projective modules

M-injective [2]. While R-injective modules are the same as injective modulesover R, the class of R-projective modules in the sense of Azumaya in general is larger than the class of projective R-modules.In this paper we introduce the notion of a strongly M-projective module and the associated notion of a strong M-projective cover.Next we investigate strong M-projective covers.We show that if every module possesses a strong M-projective cover then R/9.I(M) is (left) perfect, where I[(M) is the annihilator of M. If R/I(M) is perfect, we show that every R-module A with tM(A) 0 possesses a strong M-projective cover, where tM(A) {x A If(x) 0 for allf Hom (A, m)}.Another application of the ideas here is the result that if I(M) 0, then an R-module B is strongly M-projective iff B is projective.In particular if R is (left) perfect and 9.I(M) 0, then an R-module B is M-projective iffB is actually projective.Since [(R) 0, we can regard this result as a generalization of the "known" result that when R is perfect an R-module is R-projective iff it is projective.It will be interesting to characterise the rings with the property that R-projective modules are the same as the projective modules over R. PreliminariesThroughout this paper R denotes a ring with -0, R-mod the category of unital left modules.All the modules we deal with are unital left modules.M denotes a fixed object in R-rood.We recall briefly the concepts of Mprojective and M-injective modules introduced by G. Azumaya and state two results due to him [1].DEVINITION }.1.A module P is called M-projective if given any eipmorphism b: M -, N and any f: P -, N, there exists a 9: P --* M such that b 9 f.