Let S be an associative ring with 1, A be a unitary right S-module, A is said to be self-small, if the image of any homomorphism \(A\to \sum^{\oplus}_{i\in {\mathfrak M}}A_ i\), \(A_ i\cong A\), \(i\in {\mathfrak M}\) is contained in the sum of a finite number \(A_ i\) for any set \({\mathfrak M}\). An S-module G is said to be A-free (A-projective) if \(G\cong \sum^{\oplus}_{{\mathfrak M}}A\) (G is a direct summand of \(\sum^{\oplus}_{{\mathfrak M}}A)\). G is said to be locally A-projective if any finite set of elements of G can be embedded in an A-projective direct summand of G. Let \(R=End_ sA\), \(H=Hom_ s(A,-)\), \(T=-\otimes_ RA\), \({\mathcal P}_ R\) be the category of all projective right R-modules, \({\mathcal P}_ A\) be the category of all submodules G of A-projective S- modules with \(G=tr_ GA.\) Theorem 1.1. Let A be a self-small S-module and a flat R-module. Then the following conditions are equivalent: 1) IA\(\neq A\), for any proper right ideal I of R; 2) The functors H and T define an equivalence of \({\mathcal P}_ A\) and \({\mathcal P}_ R\). Theorem 1.2. If R is discrete in the finite topology and A is a flat R-module, then the following conditions are equivalent: 1) IA\(\neq A\) for any proper right ideal I of R; 2) H and T define an equivalence of the category of all submodules G of locally A- projective S-modules with \(tr_ GA=G\) and the category of all submodules of locally projective right R-modules. Corollary 1.1. Let A be a self- small S-module and a flat R-module. The following statements are equivalent: 1) R is a right hereditary ring a ring of free right ideals); 2) IA\(\neq A\) for any proper right ideal I of R and any submodule \(G=tr_ GA\) of an A-projective (A-free) S-module is A-projective (A-free). In {\S}2 of this article \(S={\mathbb{Z}}\), \(R=End_{{\mathbb{Z}}}A\). For an abelian group G \(tr_ GA=G\) if and only if there is an epimorphism \(\sum^{\oplus}_{{\mathfrak M}}A\to G\). The smallest \({\mathcal M}\) of this kind is said to be the A-rank of G. Theorem 2.1. The following conditions on a reduced torsion-free abelian group A with noetherian ring R are equivalent: 1) R is right hereditary and for any subgroup \(V\cong A\) the group A/V is periodic; 2) if B is a subgroup of finite A-rank of a locally A-projective group G, then the pure envelope \(B_*\) of B is a direct summand of G. Corollary 2.2. Let A be a torsion-free abelian group wth a hereditary noetherian semiprime ring R and A/V be periodic for any subgroup \(V\cong A\). Then if B is a pure subgroup of a locally A- projective group G such that \(tr_ BA=B\) and the ring of endomorphisms E(B) is discrete in the finite topology, then B is a direct summand of G. Corollary 2.3. The following statements on a reduced torsion-free abelian group A of finite rank are equivalent: 1) R is right hereditary; 2) A is endoflat and if B is a pure subgroup of finite rank of a locally A- projective group G and \(tr_ BA=B\), then B is a direct summand of G; 3) If B is a subgroup of finite rank of a locally A-projective group G and \(tr_ BA=B\), then \(B_*\) is a direct summand of G.