From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3956 Path: news.gmane.org!not-for-mail From: wlawvere@buffalo.edu Newsgroups: gmane.science.mathematics.categories Subject: Re: Ideal Theory 101 [was: is 0 prime?] Date: Thu, 04 Oct 2007 20:47:32 -0400 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain X-Trace: ger.gmane.org 1241019626 11079 80.91.229.2 (29 Apr 2009 15:40:26 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:40:26 +0000 (UTC) To: categories list Original-X-From: rrosebru@mta.ca Sat Oct 6 23:19:16 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 06 Oct 2007 23:19:16 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IeLUw-00034C-8q for categories-list@mta.ca; Sat, 06 Oct 2007 23:04:02 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 13 Original-Lines: 89 Xref: news.gmane.org gmane.science.mathematics.categories:3956 Archived-At: The awesome nature of Sup cannot be the reason why the Kummer functor exists, since it is merely used for=20 recording the result. The functor is "caused" rather=20 by an internal feature of the domain category C of=20 commutative rings: The category of quotient objects of any given R has a binary operation * that is neither sup nor inf even though in principle it can be=20 expressed as a combination of limits and colimits. We can call it R/ab=3DR/a *R/b but how does the=20 operation * specialize to C concretely ? Bill Quoting Jeff Egger : > --- 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 > > > >=20