On ?-semiperfect modules

Abstract

A submodule N of a module M is -small in M if N+XM for any proper submodule X of M with M/X singular. A projective -cover of a module M is a projective module P with an epimorphism to M whose kernel is -small in P. A module M is called -semiperfect if every factor module of M has a projective -cover. In this paper, we prove various properties, including a structure theorem and several characterizations, for -semiperfect modules. Our proofs can be adapted to generalize several results of Mares [8] and Nicholson [11] from projective semiperfect modules to arbitrary semiperfect modules.