From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6366 Path: news.gmane.org!not-for-mail From: Michael Barr Newsgroups: gmane.science.mathematics.categories Subject: Re: Terminological question, and more Date: Fri, 5 Nov 2010 07:51:38 -0500 (EST) Message-ID: References: Reply-To: Michael Barr NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: dough.gmane.org 1289048257 11025 80.91.229.12 (6 Nov 2010 12:57:37 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 6 Nov 2010 12:57:37 +0000 (UTC) Cc: categories To: "Fred E.J. Linton" Original-X-From: majordomo@mlist.mta.ca Sat Nov 06 13:57:26 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.114]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1PEiKj-0007x0-8X for gsmc-categories@m.gmane.org; Sat, 06 Nov 2010 13:57:25 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:44564) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1PEiK9-0005xp-RJ; Sat, 06 Nov 2010 09:56:49 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1PEiK6-0007ew-4b for categories-list@mlist.mta.ca; Sat, 06 Nov 2010 09:56:46 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6366 Archived-At: I don't know any name for the first question. As for the second, I think the conclusion is correct, but the reasoning is not. The argument that any automorphism of R is the identity does not depend on multiplicative inverses. But any automorphism of a transcendence basis will extend to a homomorphism of R into C. This takes place even in the category of fields. As a side comment (and the context in which I learned this), the German translation of Pontrjagin's Topological Groups contains a chapter on topological fields that was omitted in the English translation (unless it has been added in the last 50 years). In that chapter is a flawed argument for the theorem that the only division algebras containing R are R, C, and H. The proof is flawed because the hypothesis did not assume, but the proof used, that R is in the center of H. C.T. Yang eventually came up with the above argument and showed that whichever copy of R in C you used, you got non-isomorphic quaternions! Of course, all versions of C are isomorphic since it is always the algebraic closure of R. I might add that, in the European style, "field" did not included commutativity. So H was called a field. Michael On Wed, 3 Nov 2010, Fred E.J. Linton wrote: > I've been asked for "... the name given in an arbitrary category > to an object A for which every mono B----->A is an isomorphism." > > I'm stumped. Any ideas? > > I've also been asked to comment on whether "the inclusion Q >-------> R of > the ring of rational numbers into that of real ones is a bimorphism, in the > category Rng of rings with units and units preserving ring homomorphisms." > > Reflexively I think: monic, yes; epic, no, as permuting any two independent > transcendentals should extend to a non-identity automorphism of R over Q. > > Am I missing something here? > > TIA; and cheers, -- Fred > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]