A note on isomorphism of endomorphism algebras of modules over valuation domains

Abstract

Sufficient conditions are given in this note to determine when an isomorphism between endomorphism algebras of torsion-free modules over a valuation domain induces an isomorphism between these modules. The main results are contained in Theorems 7 and 10 and are as follows: If M and N are separable, torsion-free R-modules over a valuation domain R such that M is homogeneous of type I and N homogeneous of type J with End R M ~ End R N, then, there is a homogeneous separable torsion-free R-module U such that M ~ U | I and N ~ U | J. In particular, if I ~ J, then M ~ N. Throughout the paper R denotes a communatite valuation domain with unit and Q its field of quotients; all modules will be unital torsion-free R-modules. Some of the stated results are true in a more general setting (such as modules over any domains etc.) which may be checked by examining the proofs. For terminology and supporting results on modules over valuation domains we will rely on [1]. Recall that valuation domains are characterized by the property that, for every q e Q either q ~ R or 1/q ~ R. Every rank one torsion-free R-module is isomorphic either to Q or to an ideal of R and our results that involve ideals of R will often include the case of Q as well, without a special notice. A torsion-free module is separable if every finite subset of its elements is contained in a finite rank completely decomposable direct summand or, equivalently, if the purification (a). of each of its elements a is a direct summand ([1], Lemma XIV 2.6). Our results are modeled after the corresponding results for abelian groups to be found