Coefficient assignability and a block decomposition for systems over rings

We characterize the class of commutative rings R such that, for any linear system (A,B) with coefficients in R, one can extract the reachable part of the system, in a way similar to the classical Kalman controllability decomposition for systems over fields. The notion of strong CA ring is introduced as the class of commutative rings over which any system verifies a strong form of the coefficient assignability property. It is shown that the class of strong CA rings lies strictly between the classes of rings with strong versions of the known pole assignability (PA) and feedback cyclization (FC) properties, defined only for reachable systems. We prove that for UCU rings, for example polynomials with coefficients in a field, the usual PA, CA and FC properties are equivalent to the corresponding strong forms of these properties, in particular C[y] is a strong CA ring.