Re: Terminological question, and more

["George Janelidze" <janelg@telkomsa.net>]

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" <fejlinton@usa.net>
To: "categories" <categories@mta.ca>
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



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