A Baer-Kaplansky theorem for modules over principal ideal domains

Abstract

We will prove that if $G$ and $H$ are modules over a principal ideal domain $R$ such that the endomorphism rings ${\mathrm{End}}_R(R\oplus G)$ and ${\mathrm{End}}_R(R\oplus H)$ are isomorphic then $G\cong H$. Conversely, if $R$ is a Dedekind domain such that two $R$-modules $G$ and $H$ are isomorphic whenever the rings ${\mathrm{End}}_R(R\oplus G)$ and ${\mathrm{End}}_R(R\oplus H)$ are isomorphic then $R$ is a PID.

preprint version; the final version is accepted by JCA