A note on isomorphism of endomorphism algebras of modules over valuation domains (Q1097314)

The author proves that the center of the endomorphism ring of a separable torsion free module is a rank one torsion free module. The properties of \(M^{\#}\) are utilised to prove the following main results: If M and N are separable, torsion free R-modules over a valuation domain R such that M is homogeneous of \(type\quad I\) and N homogeneous of \(type\quad J\) with \(End_ RM\cong End_ RN\), then there is a homogeneous separable torsion free R-module U such that \(M\cong U\otimes I\) and \(N\cong U\otimes J\). In particular, if \(I\cong J\), then \(M\cong N\).