Rosenbrock's theorem for systems over von Neumann regular rings (Q492767)

Let \((A,B)\) be a linear system over a commutative ring \(R\), that is, \(A\) and \(B\) are matrices of size \(n\times n\) and \(n\times m\), respectively, for some positive integers \(m\) and \(n\). Let \(R[s]\) be the polynomial ring over \(R\) in the indeterminate \(s\). I quote: ``This paper deals with the following invariant factor assignment problem: \((A,B,M)\): Given \((A,B)\), find \(F\) such that \(M\sim sI-(A+BF)\), where \(\sim\) denotes equivalence over \(R[s]\), i.e. we look for \(n\times n\) invertible matrices \(P(s, Q)s\) in \(R[s]\) such that \(M=P(s(sI-A(+BF)Q(s))\).'' An obvious necessary condition for solvability of the problem \((A,B,M)\) is that the leading coefficient of det \(M\) is invertible in \(R\). A matrix \(M\) with this property is called in the paper an `admissible structure' for the system \((A,B)\). The authors characterize commutative von Neumann regular rings by the following property: every problem \((A,B,M)\) (with \(M\) an admissible structure) is solvable over \(R\) if and only it has a solution modulo each maximal ideal. Thus they reduce this problem over a von Neumann regular ring to the same problem over a field. This reduction allows them to generalize Rosenbrock's pole assignment theorem to von Neumann regular rings. For a von Neumann regular ring \(R\), the solvability of a problems \((A,B,M)\) as above is a local-global property: indeed, for each maximal ideal \(\mathfrak M\) of \(R\), there is a canonical isomorphism \(R_{\mathfrak M}\cong R/\mathfrak M\), where \(R_{\mathfrak M}\) is the localization of \(R\) at the maximal ideal \(\mathfrak M\).