?-M-small and ?-Harada modules

Abstract

Let M be a right R-module and N is an element of sigma[M]. A submodule K of N is called delta-M-small if, whenever N = K + X with NIX M-singular, we have N = X. N is called a delta-M-small module if N congruent to K, K is delta-M-small in L for some K, L E a[M]. In this article, we prove that if M is a finitely generated self-projective generator in a[M], then M is a Noetherian QF-module if and only if every module in sigma[M] is a direct sum of a projective module in a[M] and a delta-M-small module. As a generalization of a Harada module, a module M is called a delta-Harada module if every injective module in sigma[M] is delta(M)-lifting. Some properties of delta-Harada modules are investigated and a characterization of a Harada module is also obtained.