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Abstract. We investigate to what extent an abelian group G is determined by the homomorphism groups Hom (G, B) where B is chosen from a set X of abelian groups. In particular, we address Problem 34 in Professor Fuchs' book which asks if X can be chosen in such a way that the homomorphism groups determine G up to isomorphism. We show that there is a negative answer to this question. On the other hand, there is a set X which determines the torsion-free groups of finite rank up to quasi-isomorphism.

A contemporary approach to studying commutative rings is to take a given theorem for abelian groups, then interpret the statement for a ring R. The validity of the new statement for R usually imposes some w x restrictions upon the structure of R. Warfield 32 showed that for torsion-free abelian groups A and B, when A has rank 1, the natural map RFor the remainder of the text, R will denote a noetherian domain. In Section 1 we consider weakly solvable domains and show that R is weakly solvable precisely when each rank 1 module is flat as a module over its Ž endomorphism ring. This in turn is equivalent to R being stable i.e., each . ideal of R is projective over its ring of endomorphisms . In Section 2 we show that stability is equivalent to being solvable as well.Several authors have considered the generalization of Warfield's dualw x wx ity result to noetherian domains 26, 15, 16, 9 . The terminology of 9 states that an integral domain R is a Warfield domain if for any rank 1 module A, the torsion-free modules B of finite rank satisfying 343

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