Pole assignability in polynomial rings, power series rings, and Prüfer domains

Abstract

The topics treated in this paper have their origins in the area of algebraic systems theory. However, the paper itself should be classified as pure commutative algebra and we shall present it as such in the body of the text. Still, it is appropriate to give a brief paragraph of motivation. If a physical system is governed by a pair (F, G) of matrices, then the stability of the system can be determined by examining the eigenvalues of the matrix F. If the system is unstable, a “feedback” matrix K can sometimes be employed in such a way that the eigenvalues of the matrix F+ GK measure the stability of the (modified) system (F+ GK, G). In this manner, an unstable system can be rendered stable. The pole assignability problem over commutative rings is one method of attacking the problem of finding such matrices K. The paper is divided into four sections. Section 1 is given over almost entirely to defining the properties in which we shall be interested. It concludes with a theorem about residuating and lifting the properties. Section 2 is concerned with the preservation of the properties under polynomial ring and power series ring formation. Section 3 is concerned with “feedback cyclization,” a strong form of pole assignability. Section 4 is concerned with pole assignability over Priifer domains. A complete elaboration of our results must await the introduction of the necessary terminology. For now, we mention the following results in somewhat vague language. If R is a zero-dimensional ring, then the pole assignability problem is solvable in R[X] and in R[ [Xl]. From a ringtheoretic standpoint, almost any class of commutative rings contains members for which the feedback cyclization problem is solvable. On the other hand, if R is a ring with 1 in its stable range, then the feedback cyclization problem is solvable in R if and only if a certain nice matricial property