Cyclic accessibility of reachable states characterizes von Neumann regular rings

The class of commutative von Neumann regular rings is characterized by a generalization of the feedback cyclization property to non-necessarily reachable systems: for any system (A,B), there exist a matrix K and a vector u such that (A,B) and the single-input system (A+BK,Bu) have the same submodule of reachable states. An explicit algorithm is presented to obtain K,u for a given system (A,B).