Simple necessary and sufficient conditions for the solvability of many control problems for linear systems over a field are based on the equivalence between the (A, B)-invariance property and the (A + BF)-invariance, or feedback invariance, property. For systems over a ring, this equivalence is no longer true and many results of the geometric control theory cannot be extended. In this paper we will present new, algorithmically checkable characterizations of the (A + BF)-invariance property for systems defined over a principal ideal domain and a new solvability condition for the disturbance rejection problem.