Comparing First Order Theories of Modules over Group Rings

We consider R‐torsionfree modules over group rings RG, where R is a Dedekind domain and G is a finite group. We compare the (first order) theory T of al these modules and the theory T₀ of the finitely generated ones (so of RG‐lattices). It is easy to realize that they are equal iff R is a field. The obstruction is the existence of R‐divisible R‐torsionfree RG‐modules. Accordingly we consider R‐reduced R‐torsionfree RG‐modules for a local R. We show that the key conditions ensuring that their theory equals T₀ are: (1) RG‐lattices have a finite representation type; (2) each attice over the completion R?G is isomorphic to the completion of some RG‐lattice.Some related questions are discussed.