On Kalman models over a commutative ring

There is a good notion of rational functions with coefficients in a commutative ring. Using this notion, we easily obtain a neat generalization of Chapter 10 of the classical book by Kalman et al. to linear systems over an arbitrary commutative ring. The generalizations certainly exist already. However, we believe that the approach we present is more natural and straightforward. In this note we would like to show that the classical results of Kalman given in [Ch. 10, 5] can be generalized with absolutely no difficulty to the case when linear systems are defined over a commutative ring. For other generalizations we refer to [2, 3, 4, 6, 7]. Throughout, A is a commutative ring (with a unit of course, but not necessarily Noetherian), s an indeterminate, m an input number and p an output number. By a monic polynomial we shall understand a one whose leading coefficient is an invetible element of A. Obviously monic polynomials form a multiplicative subset in A[s]. The corresponding localization of A[s], denoted by A(s), is called the ring of rational functions with coefficients in A. Thus, by definition, A(s) consists of fractions of the form f/g, where f is an arbitrary polynomial and g a monic polynomial. We remark that a monic polynomial can not be a zero divisor. Hence, f1/g1 = f2/g2 if and only if f1g2 = f2g1. This implies, in particular, that the canonical homomorphism A[s]→ A(s), f 7→ f/1 is an embedding, and we shall identify A[s] with its image under this embedding. A rational function f/g is called proper if deg(f) ≤ deg(g). It is easily seen that proper rational functions form a ring, and we shall denote it by O. Clearly, we have A(s) = ∩n≥0sO. Notice that the Euclidean algorithm holds; namely, if f and g are polynomials and if g is monic, then there exists a unique pair of polynomials q and r such that f = qg + r and deg(r) < deg(g). (We put deg(0) = −∞.) Consequently, we have a fundamental relation A(s) = A[s]⊕ s−1O. Given a rational function f , define its residue Res(f) as the coefficient at s−1 in the representation of f as a series in A((s−1)). The map Res : A(s) → A is A-linear and vanishes on A[s]. Therefore it determines a canonical A-linear map A(s)/A[s] → A, which will be denoted by Res again.