Assignable polynomials to linear systems over von Neumann regular rings

Given a system (A,B)(A,B) over a commutative von Neumann regular ring R, it is proved that there exist a matrix F and a vector u   such that the single-input system (A+BF,Bu)(A+BF,Bu) and the original system (A,B)(A,B) have the same module of reachable states and the same set of polynomials assignable by state feedback. Moreover, there is a bijection between reachable states and assignable polynomials, in the form of a certain isomorphism of R-modules, and the existence of this isomorphism for all systems actually characterizes von Neumann regular rings. Finally, the set of assignable polynomials to a single-input system is completely described for arbitrary commutative rings, which in the case of von Neumann regular rings completes the study of assignable polynomials to multi-input systems.