Further results on (A+BF)invariance for systems over a principal ideal domain

It is well known that (A-B) invariance and (A+BF) invariance properties are not equivalent when we are working on systems over a ring. This fact implies to impose restrictive conditions on a system defined over a ring to extend some classical results of the geometric control theory. We present here some new characterizations of the (A+BF) invariance which can be useful for the improvement of the results given by Conte and Perdon concerning the disturbance rejection problem or the decoupling problem via static feedback control laws.