Modules over domains large in a complete discrete valuation ring

We consider a class of domains R containing a maximal ideal N such that R i s not complete with respect to the n-adic topology, but T = R-N is a comple te DVR. Such domains are called T-large because of the way to construct the m. We characterize a T-large domain R to be of the form R = T boolean AND V , where V is a mildly restricted valuation domain of Q, the field of fracti ons of T, We show that the completion ir of V has infinite rank as a V-modu le. We investigate finite rank torsion-free modules M over a T-large domain R which are Hausdorff in the N-adic topology. Making use of known results on V-modules, we obtain the following results: there exist indecomposable t orsion-free Hausdorff R-modules of any fixed rank n; every cotorsion-free H ausdorff R-algebra of rank n is the endomorphism algebra of a torsion-free module of rank 3n; the Krull-Schmidt theorem fails, that is, there exist fi nite rank torsion-free Hausdorff R-modules which admit nonisomorphic decomp ositions into indecomposable summands.