Up to now the use of geometric methods in the study of disturbance dec
oupling problems (DDPs) for systems over a ring has provided only nece
ssary conditions for the existence of solutions. In this paper we stud
y such problems, considering separately the case in which only static
feedback solutions are allowed, and the one in which dynamic feedback
solutions are admitted. In the first case, we give a complete geometri
c characterization of the solvability conditions of such problems for
injective systems with coefficients in a commutative ring. practical p
rocedures for testing the solvability conditions and for constructing
solutions, if any exist, are given in the case of systems with coeffic
ients in a principal ideal domain (PID). In the second case, we give a
complete geometric characterization of the solvability conditions for
systems with coefficients in a PID.