Page 116 - Matrices theory and applications

6.1. Rings, Principal Ideal Domains
This property seems natural because it is shared by all the rings encoun-
tered in number theory. But the ring of entire holomorphic functions is not
Noetherian: Just take for a n the function
n

Proof z → k=1 (z − k) −1 sin 2πz. 99
Let A be a principal ideal domain and let (I j ) j≥0 be a nondecreasing
sequence of ideals in A.Let I be their union. This sequence is nondecreasing
under inclusion, so that I is an ideal. Let a be a generator: I =(a). Then
a belongs to one of the ideals, say a ∈I k . Hence I⊂ I k , which implies
I j = I for j ≥ k.
We remark that the proof works with slight changes if we know that
every ideal in A is spanned by a finite set. For example, the ring of poly-
nomials over a Noetherian ring is itself Noetherian: ZZ[X]and k[X, Y ]are
Noetherian rings.
The principal ideal domains are also factorial (a short term for unique
factorization domain): Every element of A admits a factorization consist-
ing of prime factors. This factorization is unique up to ambiguities, which
may be of three types: the order of factors, the presence of invertible ele-
ments, and the replacement of factors by associated ones. This property is
fundamental to the arithmetic in A.

6.1.2 Euclidean Domains

Definition 6.1.3 A Euclidean domain is a ring A endowed with a map
N : A → IN such that for every a, b ∈ A with b =0, there exists a unique
pair (q, r) ∈ A × A such that a = qb + r with N(r) <N(b) (Euclidean
division).

A special case of Euclidean division occurs when b divides a.Then r =0
and we conclude that N(b) >N(0) for every b =0.
Classical examples of Euclidean domains are the ring of the rational
integers ZZ,with N(a)= |a|, the ring k[X] of polynomials over a field
√
, and the ring of Gaussian integers ZZ[ −1], with
k,with N(P)= 2 deg P 1
2
N(z)= |z| .Observe that if b is nonzero, the Euclidean division of b by
itself shows that N(b) is positive. The function N is often called a norm,
though it does not resemble the norm on a real or complex vector space. In
practice, one may define N(0) in a consistent way by 0 if b =0=⇒ N(b) > 0
√
(case of ZZ and ZZ[ −1]), and by −∞ otherwise (case of k[X]). With that

1
One may take either N(P )= 1 + deg P if P is nonzero, and N(0) = 0.