The block decoupling problem for systems over a ring

The decoupling problem by static-state feedback for a system with coefficients in a ring is investigated using a geometric approach. The notion of a precontrollability submodule, which captures the geometric aspects of the notion of controllability subset of the state space in the present context, is introduced and it is used for stating solvability conditions for the decoupling problem for a system with coefficients in a Noetherian ring. This result extends and improves previous results concerning systems with coefficients in a principal ideal domain and it can, in particular, be used for dealing with decoupling problem involving delay-differential systems.