An extension of Zassenhaus' theorem on endomorphism rings

Let R be a ring with identity such that R+, the additive group of R, is torsion-free. If there is some R-module M such that R subset of M subset of QR (= Q circle times(Z) R) and End(Z)(M) = R, we call R a Zassenhaus ring. Hans Zassenhaus showed in 1967 that whenever R+ is free of finite rank, then R is a Zassenhaus ring. We will show that if R+ is free of countable rank and each element of R is algebraic over Q, then R is a Zassenhaus ring. We will give an example showing that this restriction on R is needed. Moreover, we will show that a ring due to A. L. S. Corner, answering Kaplansky's test problems in the negative for torsion-free abelian groups, is a Zassenhaus ring.