On the pole assignability property over commutative rings

Abstract

This paper is concerned with the pole assignability property in commutative rings. Specifically, a commutative ring R has the pole assignability property iff given an n-dimensional reachable system ( F, G) over R and ring elements r 1,…, r n ϵ R, there exists a matrix K such that the characteristic polynomial of the matrix F+ GK is ( X− r 1) ∣ ( X− r n ). The principal theorem of this paper is Theorem 3: Let R be a commutative ring with the property that all rank one projective R-modules are free. Then R has the pole assignability property iff given a reachable system ( F, G) there is a unimodular vector in the image of G.