On the regulator problem for linear systems over rings and algebras

The regulator problem is solvable for a linear dynamical system Σ \Sigma if and only if Σ \Sigma is both pole assignable and state estimable. In this case, Σ \Sigma is a canonical system (i.e., reachable and observable). When the ring R R is a field or a Noetherian total ring of fractions the converse is true. Commutative rings which have the property that the regulator problem is solvable for every canonical system (RP-rings) are characterized as the class of rings where every observable system is state estimable (SE-rings), and this class is shown to be equal to the class of rings where every reachable system is pole-assignable (PA-rings) and the dual of a canonical system is also canonical (DP-rings).