Let A be an abelian group.
An abelian group G is locally A-projective if every finite subset of G is contained in a
direct summand P of G which is isomorphic to a direct summand of ⊕IA for some
index-set I. Locally A-projective groups are discussed without the usual
assumption that the endomorphism ring of A is hereditary, a setting in which
virtually nothing is known about these groups. The results of this paper
generalize structure theorems for homogeneous separable torsion-free groups
and locally free modules over principal ideal domains. Furthermore, it is
shown that the conditions on A imposed in this paper cannot be relaxed, in
general.
