Locally projective and locally injective modules II

الملخص EN

The object of this paper is to study the locally projective and locally injective modules.

Specifically, this paper is a continuation of study of locally projective and locally injective modules, where a new description of locally projective and locally injective modules is obtained.

The main obtained results include: 1) Let PR be a module.

The following conditions are equivalent: (1) P is a locally projective module.

(2) Tot[M,P] = ∇[M, P] for all M e mod - R.

(3) Tot[P,M] = ∇ [P,M] for all M e mod - R.

(4) [P, M ] is a ∇ — semipotent for all M e mod — R .

(5) [M, P ] is a ∇ — semipotent for all M e mod — R .

2) Let QR be a module.

The following conditions are equivalent: (1) Q is a locally injective module.

(2) Tot[Q, N] = Δ[Q, N] for all N e mod — R.

(3) Tot[N, Q] = Δ[N, Q] for all N e mod — R.

(4) [Q, N] is a Δ — semipotent for all N e mod — R .

(5) [ N, Q] is a Δ — semipotent for all N e mod — R .

3) The following conditions are equivalent for any ring R : (1) Every moduleM e mod — R with EM is a Δ — semipotent ring, is injective.

(2) R — is a semi-simple Artinian ring.

(3) Every module M e mod — R with EM is a ∇ — semipotent ring, is projective.