Re: Terminological question, and more

[Michael Barr <barr@math.mcgill.ca>]

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
>
>


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