298 8. EXTENSIONSTO GENERAL SETTINGS

So for any f, g ∈ B[x],(8.1.67) implies that

a(f +bg)V (f, bg) = a(f ) . (8.1.68)

Therefore b ∈ I, by deﬁnition. On the other hand, b is the product of elements in B \ J, and hence

itself belongs to B \ J. In particular, b/∈ I ⊆ J. This contradiction shows that I = B.

2

8.2 COPRIME FACTORIZATIONS FOR DISTRIBUTED

SYSTEMS

Suppose R is a commutative domain with identity, and let F denote the associated ﬁeld of fractions.

The previous section was concerned with the question of when a matrix in M(F) has a right or

left-coprime factorization, and when the existence of one implies the existence of the other. It was

shown that the notion of a Hermite ring plays an important role in answering the above questions.

In particular, if R is a Hermite ring, then a matrix in M(F) has an r.c.f if and only if it has an l.c.f.

However, aside from some general and abstract results, very little was said about the existence of

coprime factorizations. The objective of this section is to study this issue in more detail for the case

of linear distributed systems, both continuous-time as well as discrete-time. This is done by ﬁrst

proving a general theorem about the existence of coprime factorizations, and then specializing this

theorem to the two special cases mentioned above. As a consequence, we shall obtain two very large

classes of transfer matrices whose elements all have both r.c.f ’s as weil as l.c.f.’s.

The main result of this section is given next.

Theorem 8.2.1 Suppose I is a subset of R satisfying two conditions:

(I1) I is a saturated multiplicative system;

(I2) If a ∈ I and b ∈ R, then the ideal in R generated by a and b is principal.

Under these conditions, every matrix G ∈ M(I

−1

R) has both an r.c.f. as well as an l.c.f.

Remarks 8.2.2 1) Recall from Section A.2 that a subset I of R is said to be multiplicative if a, b ∈ I

implies that ab ∈ I.Itissaturated if a ∈ I and b ∈ R divides a together imply that b ∈ I.

2) By induction, one can show that (I2) is equivalent to the following statement: (I2’) If a ∈ I

and b

1

,...,b

n

∈ R, then the ideal in R generated by a, b

1

,...,b

n

is principal. Let d be a generator

of this principal ideal. Then one can show quite easily that d is a greatest common divisor (g.c.d.)

of the set {a, b

1

,...,b

n

}.

3) Recall from Section A.2 that I

−1

R denotes the ring consisting of all fractions of the form

a/b, where a ∈ R, and b ∈ I. One may assume without loss of generality that 1 ∈ I. By identifying

a ∈ R with a/1, one can imbed R as a subring of I

−1

R.

The proof of Theorem 8.2.1 is facilitated by some preliminary results.

Lemma 8.2.3 Suppose a ∈ I,b

1

,...,b

n

∈ R, and let d be a g.c.d. of the set {a,b

1

,...,b

n

}. Then

there exists a matrix U ∈ U(R) such that |U|=d and [a, b

1

,...,b

n

]

is the ﬁrst column of U .

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