FUNDAMENTA MATHEMATICAE 194 (2007) An extension of Zassenhaus’ theorem on endomorphism rings by Manfred Dugas (Waco, TX) and R¨ udiger G¨ obel (Essen) Abstract. 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 ⊆ M ⊆ QR (= Q ⊗ 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. 1. Introduction. In 1963 A. L. S. Corner [5] proved his celebrated result that any countable, torsion-free, reduced ring is the endomorphism ring of a countable torsion-free, reduced abelian group. Moreover, he was able to show that if R is such a ring of finite rank n, then there exists an abelian group A such that End(A)= R and A has rank 2n. He also gave an example of a ring R of rank n that is not the endomorphism ring of any group of rank less than 2n. This makes it natural to ask for which torsion-free rings R of rank n there exists an abelian group A of the same rank n such that End(A)= R. An answer was obtained by H. Zassenhaus [15] in 1967: If R is a ring with identity such that R + , the additive group of R, is free of finite rank, then there exists an R-module M such that R ⊆ M ⊆ QR and End(M )= R. Of course, the first clause means that R and M have the same rank. In 1968, M. C. R. Butler [4], using similar techniques to those in [15], extended this result to finite rank rings R such that R p , the localization of 2000 Mathematics Subject Classification : Primary 20K20, 20K30; Secondary 16S60, 16W20. Key words and phrases : endomorphism rings, Zassenhaus’ theorem. Support through a sabbatical leave granted by Baylor University in the spring of 2006 and the hospitality of the Universit¨at Duisburg-Essen are gratefully acknowledged. This work is supported by the project No. I-706-54.6/2001 of the German-Israeli Foundation for Scientific Research & Development. [239] c  Instytut Matematyczny PAN, 2007