EVERY COTORSION-FREE RING IS AN ENDOMORPHISM RING MANFRED DUGAS and RUDIGER GOBEL [Received 8 December 1980] ABSTRACT Some years ago A. L. S. Corner proved that every countable and cotorsion-free ring can be realized as the endomorphism ring of some torsion-free abelian group. This result has many interesting consequences for abelian groups. Using a set-theoretic axiom V k ., which follows for instance from V = L, we can drop the countability condition in Corner's theorem. 1. Introduction The famous Wedderburn theorem gives a ring-theoretic characterization of those rings which are endomorphism rings offinite-dimensionalvector spaces. This class of rings is very small and does not grow much if we allow infinite-dimensional vector spaces, as was pointed out by Anderson and Fuller [1, p. 152, Theorem 13.4, and p. 164, Exercise 13]. The endomorphism rings of abelian p-groups without elements of infinite height have also been characterized in terms of their ideal structure (see Baer [2], Liebert [32], and Pierce [34]), and it turns out that these classes of rings are also very restricted. The situation is completely different in the case of torsion-free abelian groups. The dramatic result of A. L. S. Corner on endomorphism rings shows this explicitly: every countable reduced torsion-free ring R is isomorphic to the endomor- phism ring End z A of a countable reduced and torsion-free abelian group A (cf. the preface of the book [22] by Griffith, and Fuchs [17, Vol. II, p. 231]). This theorem was published almost twenty years ago in this journal, Corner [4]. Corner's result, which attracted the interest of many mathematicians, has several important con- sequences for the existence of abelian groups with pathological direct-sum decom- positions. We refer only to Corner's countable counter-example to Kaplansky's Test Problem II, showing that G@G^M@M does not imply G ^ M [4, p. 704]. Further applications of Corner's theorem can be found in [17, Vol. II, §§91 and 110]. Subsequently Corner [7] and several other authors [10, 33] generalized this theorem (partially) to uncountable rings. All the known proofs depend on Corner's nice idea of taking subgroups of the p-adic completions of the additive group of R as possible groups with endomorphism rings isomorphic to R. In order to make the proof work for possibly uncountable rings, the rings must still satisfy suitable countability conditions; for example, Corner considers rings with a 'good basis', and 'residually controlled' rings, in [7, p. 63]. It is the aim of this paper to give a natural generalization of Corner's theorem to arbitrary cardinalities without any countability conditions, using a suitable axiom of set theory. The proof will be completely different from Corner's proof and will be Financial support for this paper was furnished by the Ministeriumfiir Wissenschaft und Forschuny des Landes Nordrhein-Westfalen under the title 'Uberabzahlbare abelsche Gruppen'. Proc. London Math. Soc. (3), 45 (1982), 319-336.