ON ZM-SEMIPERFECT MODULES

Abstract

Let τM be any preradical for σ[M] andN any module in σ[M]. N is called a τM-semiperfect module if for every submodule K of N, there is a decomposition K = A⊕B such that A is a projective direct summand of N in σ[M] andB ⊆ τM(N). In this paper we prove that any finite direct sum of τM-semiperfect modules is τM-semiperfect. It is also shown that if M is a local projective module in σ[M], then for every index set Λ, the sum M(Λ) is ZM-semiperfect in σ[M] if and only if every factor module of M(Λ) has a projective ZM-cover.