An abelian group G is said to be E-finitely generated (respectively, E-cyclic) if it is finitely generated (resp., cyclic) as a module over its endomorphism ring E. In this paper we refine a theorem of Reid [5] cUid apply the result to settle a question of some years standing concerning infinitely gen¬erated groups. We also determine the structure of torsion-free groups of finite rank for which Q ⊗zis a progenerator over Q ⊗zE