On weakly projective and weakly injective modules

Abstract

The purpose of this paper is to further the study of weakly injective and weakly projective modules as a generalization of injective and projective modules. For a locally q.f.d. module M, there exists a module K ∈ σ[M] such that K ⊕ N is weakly injective in σ[M], for any N ∈ σ[M]. Similarly, if M is projective and right perfect in σ[M], then there exists a module K ∈ σ[M] such that K ⊕ N is weakly projective in σ[M], for any N ∈ σ[M]. Consequently, over a right perfect ring every module is a direct summand of a weakly projective module. For some classes M of modules in σ[M], we study when direct sums of modules from M satisfy property P in σ[M]. In particular, we get characterizations of locally countably thick modules, a generalization of locally q.f.d. modules.