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.
Citation
Simion Breaz. "A Baer-Kaplansky theorem for modules over principal ideal domains." J. Commut. Algebra 7 (1) 1 - 7, SPRING 2015. https://doi.org/10.1216/JCA-2015-7-1-1
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