TORSION-FREE SEPARABLE ABELIAN GROUPS QUASI-PROJECTIVE OVER THEIR ENDOMORPHISM RINGS

To Professor L. Fuchs on the occasion of his sixtieth birthday. ABSTRACT. Necessary and sufficient conditions are obtained for a torsion-free separable abelian group A to be quasi-projective as an E(A)-module. It tums out that such an A is always flat as an E(A)-module. If the set T of types of rank one summands of a separable group A is countable, then A is E(A)-quasi-projective if and only if A is E(A)-projective. L. Fuchs [3] showed that all abelian p-groups, other than the groups of the form (divisible) * (bounded), are quasi-projective over their endomorphism rings. C. Vinsonhaler and W. J. Wickless have characterized in [5] the torsion-free completely decomposable abelian groups which are quasi-projective as modules over their endomorphism rings. In this paper, we generalize the results of [5] for the class of separable torsion-free abelian groups. We also correct Condition 3 in Theorem 2.1 of [5]. All the groups that we consider here are torsion-free abelian groups. We refer to the book by L. Fuchs [2] for the general notation and terminology used here. For any rank one group S or for any element x of a torsion-free group, let t(S) and t(x) denote respectively the type of S and x. A type t whose entries are either 0 or oo is said to be idempotent. For any type t, its reduced type t o is obtained from t by replacing all the finite entries of t by 0. Let T(A) denote the set of types of elements of A. A subset S of T(A) is said to be directed below if, for all tl,t 2 6 S, there exists t3 6 S such that t 3 •< t 1,t 2. The reduced inner type of A is = min { to: t • T(A)}. Note that the reduced inner type of A always exists, but it may not belong to T(A). For any subset X of A, (X) denotes the subgroup generated by X. If o•: A- • B is a map, odX denotes the restfiction of o • to X. An R-module M is said to be quasi-projective if to each R-ma