Strong feedback cyclization for systems over rings

For nonnecessarily reachable systems over a commutative ring R, we define a strong form of feedback cyclization (FC). With this natural generalization of the FC property we obtain a feedback reduced form for systems over strong FC rings (i.e. rings for which every system verifies the FC property). In the particular case of reachable single input systems, this gives the usual control canonical form of Bumby et al. [Remarks on the pole-shifting problem over rings, J. Pure Appl. Algebra 20 (1981) 113-127]. Also it is proved that a commutative ring with 1 in its stable range has the strong FC property if and only if it has the UCU property, which is the natural parallel form of the result given by Brewer et al. [Pole assignability in polynomial rings, power series rings and Prüfer domains, J. Algebra 106 (1987) 265-286] for the reachable case. Many classes of rings which were known to be FC rings are in fact strong FC rings, but there are FC rings which are not strong FC rings.