From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3945 Path: news.gmane.org!not-for-mail From: Jeff Egger Newsgroups: gmane.science.mathematics.categories Subject: Ideal Theory 101 [was: is 0 prime?] Date: Mon, 1 Oct 2007 12:28:39 -0400 (EDT) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1241019619 11016 80.91.229.2 (29 Apr 2009 15:40:19 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:40:19 +0000 (UTC) To: categories list Original-X-From: rrosebru@mta.ca Mon Oct 1 13:57:01 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 01 Oct 2007 13:57:01 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IcOVY-0005yw-GC for categories-list@mta.ca; Mon, 01 Oct 2007 13:52:40 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 2 Original-Lines: 57 Xref: news.gmane.org gmane.science.mathematics.categories:3945 Archived-At: --- Vaughan Pratt wrote: > I don't know much about ring theory, so I > could be confused about this, but I would have thought intersecting the= m > could only get you the square-free ideals.=20 This is correct; there is simply no way of getting around the fact that=20 ideals form not just a lattice but carry a quantale structure derived=20 from the ring. [See my next post and the quotation below.] =20 --- Bill Lawvere wrote: > The ideal product under discussion is a key > ingredient in a construction of unions of subspaces that takes into > account the clashes.=20 --- Vaughan Pratt wrote: > Starting from the prime > power ideals takes care of that but what's the trick for getting all th= e > ideals from just the prime ideals? The category Div was my suggestion > for that, but if there's a more standard approach in ring theory I'd be > happy to use that instead (or at least be aware of it---Div is starting > to grow on me). I'd point you to Wikipedia, only the relevant articles are somewhat=20 scattered about. Briefly, every ideal in a Noetherian ring can be=20 written as a finite intersection of _primary_ ideals, and this can=20 be made essentially unique by adding appropriate restrictions.=20 To obtain a more easily recognisable version of the Fundamental=20 Theorem of Arithmetic, it then remains to determine under what=20 circumstances a primary ideal must be a prime power. [A good=20 counter-example is Z[x,y], where the ideal (x,y^2) is primary,=20 but falls strictly between the prime ideal (x,y) and its square (x,y)^2=3D(x^2,xy,y^2).] =20 A Noetherian integral domain which does have this extra property=20 is called a Dedekind domain; examples include the ring of algebraic=20 integers w.r.t. an arbitrary number field---proving the latter result=20 (which is connected to an infamously incorrect proof of Fermat's=20 last theorem) is commonly cited as Dedekind's original motivation=20 for defining ideals. See http://en.wikipedia.org/wiki/Primary_decomposition and http://en.wikipedia.org/wiki/Dedekind_domain for details. =20 Cheers, Jeff Egger. Ask a question on any topic and get answers from real people. Go to= Yahoo! Answers and share what you know at http://ca.answers.yahoo.com