Linear systems over Noetherian rings in the behavioural approach

In this paper we will study dynamical systems over Noetherian rings. We will follow the behavioural approach. We first study two particular cases: au-tonomous systems and controllable systems. In the first case we will be able to connect autonomous systems with finitely generated systems. Moreover we will propose several different concepts of controllability and analyze how they are connected each other. Finally we will define the concept of state space module. In this way we will say that the system is realizable if its state module is finitely generated. The last part of the paper is devoted to the analysis of the properties that ensure the realizability of the system.