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 $\End_R(R\oplus G)$ and $\End_R(R\oplus H)$ 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 $\End_R(R\oplus G)$ and $\End_R(R\oplus H)$ are isomorphic, then $R$ is a PID.