TORSION FREE ABELIAN GROUPS QUASI-PROJECTIVE OVER THEIR ENDOMORPHISM RINGS

Certain classes of torsion free abelian groups which are quasi-projective as modules over their endomorphism rings are characterized. The main results concern completely decompos-able and strongly indecomposable groups. 1. Preliminaries. Abelian groups which are quasi-projective over their endomorphism rings have been characterized by Fuchs in the torsion case. His methods have been extended by Longtin to the algebraically compact and cotorsion groups [5]. In this paper, we investigate some other classes of groups with this property. Specifically: DEFINITION. A (left) module M over a ring R is quasi-projective provided the natural map HomR(M, M)—>HomR(M,MAK) is epic for every submodule K, of M. An abelian group G will be called £-quasi-projective (Eqp) pro-vided G is quasi-projective as a module over E = End (G). Henceforth, the word group will denote a'torsion free abelian group. Other notation follows Fuchs [4], in particular, t(G)- type G for any group G of rank 1. The following simple lemmas will be quite useful. LEMMA 1.1. Let G be Eqp and K a fully invariant subgroup of G. Then G/K is a quasi-projective E-module. Proof. See Proposition 2.1 in Wu and Jans [9]. LEMMA 1.2. Let G be Eqp and K a fully invariant subgroup. Then ZE, the center of E, maps onto HomE(G/K, G/K). Proof Let Π: G — » G/K be the factor map. Since G is Eqp, for every θ GHom £ (G/X,G/X), there exists a G Hom£(G,G) = ZE such that Πα = ΘU. LEMMA 1.3. Let G be Eqp and K a fully invariant subgroup such that G/K is torsion. Then if ZE is countable, G/K is bounded