A Baer-Kaplansky theorem for modules over principal ideal domains

TL;DR

The paper establishes a characterization of modules over principal ideal domains by showing that isomorphic endomorphism rings imply module isomorphism, and conversely, that this property characterizes PIDs among Dedekind domains.

Contribution

It proves a Baer-Kaplansky type theorem for modules over PIDs and characterizes PIDs via endomorphism ring isomorphisms.

Findings

Isomorphic endomorphism rings imply module isomorphism over PIDs.

The property characterizes PIDs among Dedekind domains.

Provides a new criterion for identifying PIDs based on module endomorphisms.

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.