Locally Projective and Locally Injective Modules

The object of this paper is to study the endomorphism rings of locally projective and locally injective modules. Specifically, this paper is a continuation of study of endomorphism rings of locally projective and locally injective modules to be semipotent rings. The main obtained results include: (a) Let R R be a locally projective module over a ring R , then for any M ∈mod − R the following are equivalent: (1) [M, P] is a semipotent. (2) Tot [M, P] = J[M, P] = ∇ [M, P] . (3) For any α ∈[M,P] J[M,P] there exists β∈[P,M]with 0 ≠ Im(αβ ) ⊆⊕ P . In particular, the endomorphisms ring E P of P is a semipotent ring if and only if, for any α ∈ EP/ J Ep there exists 0 ≠ β ∈ E P such that o ≠ Im(αβ ) ⊆⊕ P . (b) Let QR be a locally injective module over a ring R , then for any module N ∈mod − R the following are equivalent: (1) [Q, N] is a semipotent. (2) Tot [Q, N] = J[Q, N] = Δ[Q, N] . (3) For any α ∈ [Q,N] J[Q,N] there exists β∈[N,Q]with 0 ≠ Ker(βα ) ⊆⊕ Q. In particular, EQ is a semipotent ring if and only if, for any α ∈ EQ /J EQ there exists 0 ≠ β ∈ EQ such that 0 ≠ Ker(βα ) ⊆⊕ Q.