From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6365 Path: news.gmane.org!not-for-mail From: "George Janelidze" Newsgroups: gmane.science.mathematics.categories Subject: Re: Terminological question, and more Date: Fri, 5 Nov 2010 13:52:11 +0200 Message-ID: References: Reply-To: "George Janelidze" NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1289048196 10843 80.91.229.12 (6 Nov 2010 12:56:36 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 6 Nov 2010 12:56:36 +0000 (UTC) To: "Fred E.J. Linton" , "categories" Original-X-From: majordomo@mlist.mta.ca Sat Nov 06 13:56:32 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 1PEiJs-0007bj-7p for gsmc-categories@m.gmane.org; Sat, 06 Nov 2010 13:56:32 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:44558) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1PEiJI-0005qb-TK; Sat, 06 Nov 2010 09:55:56 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1PEiJG-0007dS-Ef for categories-list@mlist.mta.ca; Sat, 06 Nov 2010 09:55:54 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6365 Archived-At: Dear Fred, (my comment is about "more" - not about the terminological question) The group Aut(R) of ring automorphisms of R is well known to be trivial. Proof: (a) Every automorphism takes squares to squares. (b) A real number is a square if and only if it is non-negative. (c) As follows from (a) and (b), every ring automorphism of R preserves order. (d) Aut(Q) is trivial (e) As follows from (c) and (d), Aut(R) is trivial. However, you are right that Q >---> R is not an epimorphism of course. Just use the morphisms into the field C of complex numbers in the same way as you used automorphisms of R. Similarly, using the fact that every algebraic extension of fields of characteristic 0 is separable, it is easy to show that a field extension of characteristic 0 is an epimorphism in the category of commutative rings if and only if it is an isomorphism. Greetings - George ----- Original Message ----- From: "Fred E.J. Linton" To: "categories" Sent: Wednesday, November 03, 2010 11:40 PM Subject: categories: Terminological question, and more 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/ ]