Pole assignability over commutative rings. (English) Zbl 0907.93018

Kautschitsch, Hermann

Contributions to general algebra 10. Selection of lectures given at the conference on general algebra, Klagenfurt, Austria, May 29–June 1, 1997. Klagenfurt: Verlag Johannes Heyn. 215-221 (1998).

The paper commences with the well-known way to describe linear systems over a commutative ring by a pair of matrices and repeats the definitions of concepts such as reachability, pole-assignability (PA), feedback cyclization (FC) and good-contains-unimodular (GCU) in this setting. H. Kautschitsch [Contrib. General Algebra 7, 215-220 (1991; Zbl 0761.93015)] has shown that, for homomorphisms \(h: R \rightarrow R'\) with \( 1 + \text{ker}(h) \subseteq U(R)\), also the inverse images \(h^{-1} (R')\) have the FC- (GCU-) property if \(R'\) has. The necessity of the condition is also shown to be true. This yields a strategy for showing that a ring \(R\) has the FC- (GCU-) property: Look for an epimorphic image \(h(R)\) having the desired property. If \( 1+ \text{ker}(h) \subseteq U(R)\), then \(R\) has the property. This shows, for example, the (known) result that \(R[[X]]\) has FC (GCU) iff \(R\) has. It is pointed out that the same is not true for polynomial rings and the remainder of the paper is dedicated to studying these rings. Some known results are restated by using the above or related strategies, concluding with two corollaries showing that both the integers and the polynomial ring \(F[X]\) over any ordered skewfield \(F\) fail to have the FC-property.
For the entire collection see [Zbl 0889.00019].

Reviewer: F.Théron (Pretoria)

MSC:

93B25	Algebraic methods
93B55	Pole and zero placement problems
93B52	Feedback control
13B25	Polynomials over commutative rings
13A99	General commutative ring theory

Keywords:

pole assignability; commutative rings; polynomial rings; feedback cyclization