Locally Projective and Locally Injective Modules II

The object of this paper is to study the locally projective and locally injectivemodules. Specifically, this paper is a continuation of study of locally projectiveand locally injective modules, where a new description of locally projective andlocally injective modules is obtained. The main obtained results include:(a) Let R P be a module. The following conditions are equivalent:(1) P is a locally projective module.(2) Tot[M, P] = Ñ[M, P] for all M Îmod - R .(3) Tot[P,M] = Ñ[P,M] for all M Îmod - R .(4) [P,M] is a Ñ-semipotent for all M Îmod - R .(5) [M, P] is a Ñ-semipotent for all M Îmod - R .(b) Let R Q be a module. The following conditions are equivalent:(1) Q is a locally injective module.(2) Tot[Q,N] = D[Q,N] for all N Îmod - R.(3) Tot[N,Q] = D[N,Q] for all N Îmod - R.(4) [Q,N] is a D- semipotent for all N Îmod - R.(5) [N,Q] is a D- semipotent for all N Îmod - R.(c) The following conditions are equivalent for any ring R:(1) EverymoduleM Îmod - R with M E is a D- semipotent ring, is injective.(2) R- is a semi-simple Artinian ring.(3)Every module M Îmod - R with M E is aÑ-semipotent ring, is projective