# The tensor and torsion products of modules over valuation domains

## Description

In this thesis we investigate the structure of the tensor and the torsion products of modules. The problem of determining the structures of tensor and torsion products is, in general, not an easy one. Some information is known in the case of abelian groups. We generalize these known results to modules over more general rings In order to retain the pleasant properties of torsion and torsion-free modules, we consider modules over Prufer domains where it is known that a module is flat if and only if it is torsion-free, and the tensor product of torsion-free modules is also torsion-free. We concentrate especially on modules over valuation domains since these are the local Prufer domains We investigate some general properties of tensor products. We are able to determine more fully the structure of the tensor products of uniserial modules. A module U is uniserial if the R-submodules of U are totally ordered by inclusion. In particular, a standard uniserial module is a quotient of an R-submodule of the field of quotients of the valuation domain R. Standard uniserial modules are determined by the annihilators and heights of elements. In this thesis we calculate the annihilators and heights of arbitrary elements of the tensor products of standard uniserial modules. We apply these results on uniserial modules to find the projective dimensions of the product of two ideals of a valuation domain It is well known that for abelian groups, Tor(,1) can be described easily by generators and relations. We prove that Tor(,1)('R) can be described in an analogous fashion when R is a valuation domain although the proof is very different. These results are then used to study the tensor and torsion products of polyserial and separable modules