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Abstract: We characterize the class of commutative rings R such that, for any linear system with coefficients in R, one can extract the reachable part of the system, in a way similar to the classical Kalman controllability decomposition for systems over fields. The notion of strong CA ring is introduced as the class of commutative rings over which any system verifies a strong form of the coefficient assignability property. It is shown that the class of strong CA rings lies strictly between the classes of rings with strong versions of the known pole assignability (PA) and feedback cyclization (FC) properties, defined only for reachable systems. We prove that for UCU rings, for example polynomials with coefficients in a field, the usual PA, CA and FC properties are equivalent to the corresponding strong forms of these properties, in particular is a strong CA ring. [Copyright &y& Elsevier]

Abstract: We study linear systems (F, G) with coefficients in a commutative ring R with the GCU property: whenever (F, G) is controllable, there exists a vector u with Gu unimodular. Our main result is: a GCU ring is an FC2 ring (i.e. every two-dimensional controllable system over R is feedback cyclizable) if and only if R satisfies the following property: for any finitely generated ideal I and f invertible modulo I there exist a unit u and an element h with f ≡ uh 2 (mod I). [Copyright &y& Elsevier]

If ( A , B ) is a finite system over a commutative von Neumann regular ring R , the problem of searching for a matrix F such that the pencil [ s I − A − B F ] has some prescribed Smith normal form is reduced to the case where R is a field, a problem which for controllable systems is described by a well-known theorem of Rosenbrock on pole assignment [12] , and was then generalized to noncontrollable pairs [14] . It this paper, von Neumann regular rings are characterized as the class of commutative rings for which the solution of the above problem over the ring is equivalent to its solution in each residue field. [ABSTRACT FROM AUTHOR]