Since the early work of J. D. Reid, homological properties of abelian groups have been studied not only as a method for characterizing various classes of torsion-free abelian groups, but also in order to provide examples in module theory. In this technique, abelian groups are often viewed as modules over their endomorphism ring. A modification of this method, which has been developed in the theory of torsion-free abelian groups of finite rank, is to tensor a group A with Q and study the resulting group QA as a module over its quasi-endomorphism ring QE(A). The main advantage gained by this is that QA is a finitely generated module over the Artinian ring QE(A) whose QE(A)-module structure hopefully is more accessible than the E(A)module structure of A. Over the years, the validity of such an approach has been established by a series of results of which we only want to mention Reid's and Arnold's work (see [3], [9], and [10]) as well as Jonsson's description of the quasi-decomposition of torsion-free abelian groups of finite rank [8]. We refer the interested reader to [3] for more detailed information. Examples of torsion-free abelian groups of finite rank with a prescribed endomorphism ring are frequently constructed using the realization theorems of Corner [5] and Zassenhaus [11]. Corner's result shows that each ring R whose additive group is torsion-free and reduced of rank n can be realized by a torsion-free group A of rank 2n, while Zassenhaus' work yields that the rank of the group can be lowered to n if the additive group of R is a free group of rank n. Each of these methods will construct A in such a way that it is flat as an E(A)-module, from which it follows that QA is a projective QE(A)-module. Faticoni and Goeters provided a construction method in [7] which not necessarily resulted in an abelian group which is flat as a module over its endomorphism ring. In particular, they showed that a reduced subring R of a finite dimensional Q-algebra is hereditary if and only if every torsion-free abelian group A of finite rank with E(A) = R is E(A)-flat. Unfortunately, their result is only of limited use if we investigate the structure of QA as a module over the quasi-endomorphism ring of A. Our first result shows the surprising fact that the condition that QE(A) is a hereditary ring is not strong enough to guarantee that QA is a flat QE(A)-module. EXAMPLE l. There exists a torsion-free abelian group A of rank3 such that QA is not flat as a QE(A)-module although QE(A) is right and left hereditary.