The pole assignability property in polynomial rings over GCD-domains
type: Item
instance of: scholarly article
title: The pole assignability property in polynomial rings over GCD-domains
zbMATH Open document ID: 0574.13011
DOI: 10.1016/0021-8693(85)90058-4
author: James Brewer
author: William J. Heinzer
author: David C. Lantz
published in: Journal of Algebra
publication date: 1985-01-01T00:00:00Z
review text: Let R be a commutative ring with F and G matrices over R of sizes \(n\times n\) and \(n\times m\), respectively. The pair (F,G) is said to be reachable iff the R-module generated by the columns of the matrix \([G,FG,...,F^{n-1}G]\) is \(R^ n\). The ring R is said to have the PAF- property iff each reachable pair (F,G) is ''pole-assignable'' - i.e., given arbitrary elements \(r_ 1,...,r_ n\) in R, there exists an \(m\times n\) matrix K such that the characteristic polynomial of the matrix \(F+GK\) is \((x-r_ 1)...(x-r_ n)\). The main result of this paper is proposition 3: Let D be a GCD domain and X a set of indeterminates. Suppose there is a finitely generated prime ideal Q of D[X] such that D[X]/Q has an ideal I satisfying: (i) I is generated by two elements, (ii) I is not principal, and (iii) the square of I (as an ideal) is principal. Then D[X] does not have the PAF-property. As a consequence, one obtains the well-known facts that k[x,y] and \({\mathbb{Z}}[x]\) are not PAF-rings for k a field and \({\mathbb{Z}}\) the ring of integers. One question left open by the authors is the following: If V is a Noetherian valuation domain, does V[x] have the PAF-property?
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 93B55
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15B33
zbMATH DE Number: 3916430
zbMATH Keywords: pole assignability
zbMATH Keywords: polynomial ring
zbMATH Keywords: reachable pair
zbMATH Keywords: GCD domain
zbMATH Keywords: PAF- rings
MaRDI profile type: Publication
full work available at URL: 0021-8693(85)90058-4
OpenAlex ID: W2052155648
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description: scientific article; zbMATH DE number 3916430