Abstract

A module is semi-perfect iff every factor module has a projective cover. A module M=A+B (for submodules A and B) is amply supplemented iff there exists a submodule A′ (called a supplement of A) of B such M=A+A′ and A′ is minimal with this property. If B=M then M is supplemented. Kasch and Mares [1] have shown that the first and last of these conditions are equivalent for projective modules. Here it is shown that an arbitrary module is semi-perfect iff it is (amply) supplemented by supplements which have projective covers, an extension of the result of Kasch and Mares [1]. Corresponding results are obtained for F-semi-perfect modules.