Duality and the regulator problem for linear systems over commutative rings

A commutative ring with identity is called a full quotient ring if every element of R is either a unit or a zero divisor. The Duality Principle, “Reachability is dual to observability,” holds for (stationary, discrete-time) linear systems over Noetherian full quotient rings. The solvability of the pole assignment problem is equivalent to reachability for these systems, and (dually) the solvability of the state-estimation problem is equivalent to observability. To summarize these facts, we say that: The regulator problem is solvable for linear systems over Noetherian full quotient rings