An extension of Zassenhaus' theorem on endomorphism rings
Volume 194 / 2007
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 $R\subseteq M\subseteq\mathbb{Q}R\ (=\mathbb{Q}\otimes_{\mathbb{Z}}R)$ and ${\rm End}_{\mathbb{Z} }(M)=R$, 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 $\mathbb{Q}$, 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.
