Page 115 - Matrices theory and applications

6. Matrices with Entries in a Principal Ideal Domain; Jordan Reduction
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denoted by k[X], are also principal ideal domains. More generally, every
Euclidean domain is a principal ideal domain (see Proposition 6.1.3 below).
In a commutative integral domain one says that d is a greatest common
divisor (gcd)of a and b if d divides a and b, and if every common divisor
of a and b divides d. In other words, the set of common divisors of a and
b admits d as a greatest element. The gcd of a and b, whenever it exists,
is unique up to multiplication by an invertible element. We say that a
and b are coprime if all their common divisors are invertible; in that case,
gcd(a, b)= 1.
Proposition 6.1.1 In a principal ideal domain, every pair of elements has
a greatest common divisor. The gcd satisfies the B´ezout identity: For every
a, b ∈ A,there exist u, v ∈ A such that
gcd(a, b)= ua + vb.
Such u and v are coprime.
Proof
Let A be a principal ideal domain. If a, b ∈ A,the ideal I =: (a, b)
spanned by a and b, which is the set of elements of the form xa + yb,
x, y ∈ A,isprincipal: I =(d), where d =gcd(a, b). Since a, b ∈I, d divides
a and b.Furthermore, d = ua + vb because d ∈I.If c divides a and b,then
c divides ua + vb; hence divides d, which happens to be a gcd of a and b.
If m divides u and v,then md|ua + vb; hence d = smd.If d =0, one has
sm = 1, which means that m ∈ A .Thus u and v are coprime. If d =0,
∗
then a = b =0, and one may take u = v = 1, which are coprime.

Letus remark thatagcdof a and b is a generator of the ideal aA+bA.It
is thus nonunique. Every element associated to a gcd of a and b is another
gcd. In certain rings one can choose the gcd in a canonical way, such as
being positive in ZZ, or monic in k[X].
The gcd is associative: gcd(a, gcd(b, c)) = gcd(gcd(a, b),c). It is therefore
possible to speak of the gcd of an arbitrary finite subset of A.In the above
example we denote it by gcd(a, b, c). At our disposal is a generalized B´ezout
formula: There exist elements u 1 ,... ,u r ∈ A such that
gcd(a 1 ,... ,a r )= a 1 u 1 + ··· + a r u r .

Definition 6.1.2 Aring A is Noetherian if every nondecreasing (for in-
clusion) sequence of ideals is constant beyond some index: I 0 ⊂ I 1 ⊂ ··· ⊂
I m ⊂ ··· implies that there is an l such that I l = I l+1 = ··· .
Proposition 6.1.2 The principal ideal domains are Noetherian.
Observe that in the case of principal ideal domains the Noetherian property
means exactly that if a sequence a 1 ,... of elements of A is such that every
element is divisible by the next one, then there exists an index J such that
the a j ’s are pairwise associated for every j ≥ J.