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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