## The Ring is noetherian

The ring of integers is noetherian because every ideal of is generated by one element.

Proposition 2.2.10   Every ideal of the ring of integers is principal.

Proof. Suppose  is a nonzero ideal in  . Let  the least positive element of . Suppose that is any nonzero element of . Using the division algorithm, write , where  is an integer and . We have and , so our assumption that is minimal implies that , so is in the ideal generated by . Thus  is the principal ideal generated by .

Example 2.2.11   Let be the ideal of generated by and . If , with , then , since and . Also, , so .

The ring in SAGE is ZZ, which is Noetherian.

sage: ZZ.is_noetherian()
True

We create the ideal in SAGE as follows, and note that it is principal:
sage: I = ideal(12,18); I
Principal ideal (6) of Integer Ring
sage: I.is_principal()
True

We could also create as follows:
sage: ZZ.ideal(12,18)
Principal ideal (6) of Integer Ring


Proposition 2.2.7 and 2.2.10 together imply that any finitely generated abelian group is noetherian. This means that subgroups of finitely generated abelian groups are finitely generated, which provides the missing step in our proof of the structure theorem for finitely generated abelian groups.

William Stein 2012-09-24