Finite multidimensional behaviors

We study finitely generated and finite systems defined by linear partial difference equations with constant coefficients in a Noetherian ring. The two notions coincide for finite rings, and we show that all finite systems exhibit certain periodicity properties. For the particular case of systems over the ring of integers modulo m?>?1, which is important in coding theory, a method is given for determining the number of trajectories, using Gr?bner basis techniques.