On matrix inverses modulo a subspace

Given a pair of matrices (A,B) Rn×n×Rn×m with coefficients in a commutative ring we study the problem of finding a matrix F such that A+BF becomes invertible. We point out some relations between the problem of finding feedback inverses of A modulo B and the pole-shifting problem for the pair (A,B). In fact we give feedback invertibility results over a well known large class of rings related with the pole-shifting: the class of PA rings. On the other hand we also give a pointwise-global characterization. The ring of rational integers and the coordinate ring of the real unit circle are studied in some detail. Finally the problem of derivative feedback standardization of generalized linear systems is reviewed as a particular case.