An extension of Zassenhaus' theorem on endomorphism rings

Manfred Dugas, and Rüdiger Göbel. "An extension of Zassenhaus' theorem on endomorphism rings." Fundamenta Mathematicae 194.3 (2007): 239-251. <http://eudml.org/doc/282894>.

@article{ManfredDugas2007,
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.},
author = {Manfred Dugas, Rüdiger Göbel},
journal = {Fundamenta Mathematicae},
keywords = {endomorphism rings; Zassenhaus rings; free Abelian groups of countable rank; realizing rings as endomorphism rings of Abelian groups},
language = {eng},
number = {3},
pages = {239-251},
title = {An extension of Zassenhaus' theorem on endomorphism rings},
url = {http://eudml.org/doc/282894},
volume = {194},
year = {2007},
}

TY - JOUR
AU - Manfred Dugas
AU - Rüdiger Göbel
TI - An extension of Zassenhaus' theorem on endomorphism rings
JO - Fundamenta Mathematicae
PY - 2007
VL - 194
IS - 3
SP - 239
EP - 251
AB - 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.
LA - eng
KW - endomorphism rings; Zassenhaus rings; free Abelian groups of countable rank; realizing rings as endomorphism rings of Abelian groups
UR - http://eudml.org/doc/282894
ER -