The ring
in *SAGE* is `ZZ`, which is Noetherian.

sage: ZZ.is_noetherian() TrueWe create the ideal in

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