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 and , 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm194-3-2,
author = {Manfred Dugas and R\"udiger G\"obel},
title = {An extension of Zassenhaus' theorem on endomorphism rings},
journal = {Fundamenta Mathematicae},
volume = {193},
year = {2007},
pages = {239-251},
zbl = {1122.20026},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm194-3-2}
}
Manfred Dugas; Rüdiger Göbel. An extension of Zassenhaus' theorem on endomorphism rings. Fundamenta Mathematicae, Tome 193 (2007) pp. 239-251. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm194-3-2/