Decidable representations Original Research Article

We are concerned here with the decision problem for lattices over a group ring ZG where G is a finite group. We show that if G is nilpotent and, for some prime p, G has a homomorphic image H such that H is a p-group and H is cyclic of order ≥ p4, or H is C(p)2 and p is odd, or H is C(2)3 or C(4) × C(2) and p = 2, then the theory of Z G-lattices is undecidable. On the contrary, lattices over Z2C(2)2 or over some rings Z G of finite representation type have a decidable theory.