From: Boudewijn Moonen Subject: Re: abelian groups Date: Mon, 19 Mar 2001 14:15:02 +0100 Newsgroups: sci.math Summary: Finitely generated abelian groups satisfy ascending chain condition. druid wrote: > > Can someone please give a hint, I need to show that > finitely generated abelian groups satisfy the > ascending chain condition. Thank you very much. Let 0 -> A' -> A -> A'' -> 0 be a short exact sequence of abelian groups. Then show A has ACC <=> A' and A'' have ACC . In particular, any f.g. free abelian group, and any quotient of an abelian group with ACC, has ACC. Since Z, the integers, has ACC, and any f.g. abelian group is a quotient of a f.g. free abelian group, the claim follows. In fact, the same logic shows, more generally, that the f.g. modules over a noetherian ring are just the noetherian ones, i.e. those having ACC. Regards, -- Boudewijn Moonen Institut fuer Photogrammetrie der Universitaet Bonn Nussallee 15 D-53115 Bonn GERMANY e-mail: Boudewijn.Moonen@ipb.uni-bonn.de Tel.: GERMANY +49-228-732910 Fax.: GERMANY +49-228-732712