From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3938 Path: news.gmane.org!not-for-mail From: Jeff Egger Newsgroups: gmane.science.mathematics.categories Subject: Re: The division lattice as a category: is 0 prime? Date: Thu, 27 Sep 2007 17:59:41 -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 1241019615 10988 80.91.229.2 (29 Apr 2009 15:40:15 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:40:15 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Fri Sep 28 11:13:42 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 28 Sep 2007 11:13:42 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IbGQH-0000nc-4J for categories-list@mta.ca; Fri, 28 Sep 2007 11:02:29 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 62 Original-Lines: 51 Xref: news.gmane.org gmane.science.mathematics.categories:3938 Archived-At: --- Vaughan Pratt wrote: > I arrived at all this after Steve Vickers mentioned on the univalg > mailing list that ring theorists define 0 to be a prime number because > then they could define n to be prime just when the ring Z/nZ extends to > a field. =20 Um, well, for arbitrary ideals I in a commutative ring R, R/I "extends to= =20 a field" (or, in more common parlance, "is an integral domain") if and on= ly=20 if I is a prime ideal; hence the previous assertion can be simplified to=20 ring theorists define 0 to be a prime number because then they could define n to be prime just when nZ is a prime ideal.=20 which doesn't seem so unreasonable. =20 > This got me to wondering how this could be reflected in the > division lattice, which has 0 at the top without however being > considered a prime. I personally am too old to believe that 0 is a > prime, but I can see where a younger generation could be hoodwinked. And I thought that every generation since Dedekind, Krull and Noether knew that divisibility lattices are (in the general case) a red herring=20 and that it is the lattice of ideals of a ring (or its opposite, if you=20 prefer) which is really important. Surely, it makes sense to fix=20 terminology according to what does work in the general case. =20 > Even with the above understanding however I don't see how 0 can be > understood as just another ordinary prime, any more than bottom is just > another ordinary number in N_*. Although 0 can be a prime (depending on the ring under consideration),=20 it is plainly never "just another ordinary prime": there is a well-known topology on the set of prime ideals of a commutative ring which clearly distinguishes 0 from its fellows. Perhaps the answer to your original=20 question is to take (finite-valued) sheaves on this space of primes,=20 although I don't really understand your motivation. =20 Cheers, Jeff Egger. Get news delivered with the All new Yahoo! Mail. Enjoy RSS feeds r= ight on your Mail page. Start today at http://mrd.mail.yahoo.com/try_beta= ?.intl=3Dca