Locally projective and locally injective modules

الملخص EN

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 PR be a locally projective module over a ring R , then for any M e mod-R the following are equivalent: (1) [M, P] is a semipotent.

(2) Tot [M, P] = J[M, P] = V [M, P].

(3) For any ae[M,P]\J[M,P] there exists 0e[P,M\with 0 * Im(a0) c® P.

In particular, the endomorphisms ring Ep of P is a semipotent ring if and only if, for any a e Ep \ J(Ep ) there exists 0 * 0 e Ep such that o * Im(a0) e® P.

(b) Let Qr be a locally injective module over a ring R , then for any module N e mod -R the following are equivalent: (1) [Q, N] is a semipotent.

(2) Tot [Q, N] = J[Q, N] = A [Q, N].

(3) For any ae [Q, N]\J[Q, N] there exists 0e[N,Qwith 0 * Ker(0a) c® Q.

In particular, eQ is a semipotent ring if and only if, for any a e Eq \ J(Eq ) there exists 0 ^ 0 e Eq such that 0 * Ker{0a) c® Q.