A SURVEY OF THE REALIZATION AND DUALITY THEORIES OF LINEAR SYSTEMS OVER RINGS

Abstract

Important aspects of the theory of linear systems over rings are the realization problem, the descent problem and the problem of duality. A survey of these problems is made and an indication given of the relevant aspects of modern algebra. A power series is associated with a response map and a necessary and sufficient condition is given for a finite realization, viz. the rationality of the associated power series. In this way a purely algebraic characterization is obtained of finite and free realizations. In addition, classes of rings are characterized for which certain system theories hold, it being shown for example that the realization problem over an integral domain R can be reduced to a similar problem over the field of quotients of R if and only if R is a. Noetherian integral domain.