The ring in SAGE is ZZ, which is Noetherian.
sage: ZZ.is_noetherian() TrueWe 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() TrueWe 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