Comparing first order theories of modules over group rings (Q2776820)

Throughout, \(R\) is a Dedekind domain of characteristic 0 and \(G\) is a finite group. An \(RG\)-lattice is a finitely generated torsion-free \(RG\)-module. The question initially addressed by the authors is whether the theory of \(RG\)-lattices is equal to the theory of \(R\)-torsion-free \(RG\)-modules. They show that these theories are equal if and only if \(R\) is a field. So they modify the question by asking when the theory, \(T_0\), of \(RG\)-lattices is equal to the theory, \(T_1\), of \(R\)-torsion-free \(R\)-reduced \(RG\)-modules. They show that if \(R\) is local then these are equal provided that \(RG\) is of finite lattice representation type and provided that Heller's condition holds (every \(\hat{R}G\)-lattice is the completion of an \(RG\)-lattice, where \(\hat{R}\) is the completion of \(R\) at its maximal ideal). (If \(R\) is not local then \(T_0\neq T_1\).) They establish the necessity of these conditions, at least under some additional hypotheses. NEWLINENEWLINENEWLINETheir proof gives rise to the following question: if \(M\) is an \(R\)-reduced \(RG\)-module, is the pure-injective hull of \(M\) also \(R\)-reduced? They show that this is so in some particular cases, but the general question is left open.