Talk:Ring Homomorphism from Division Ring is Monomorphism or Zero Homomorphism

I believe we've already got this one covered:

Epimorphism from Division Ring to Ring

It does the same sort of job except:

a) It starts with an epimorphism (and therefore concludes with a bijection) - but the homomorphism version is equivalent;

b) It allows that it also works for a division ring, thus making it slightly stronger.

oh yes and "homorphism" may need to be corrected as to spelling.

--Matt Westwood 13:06, 20 December 2008 (UTC)

Yeah I was thinking it generalized to Division Rings, since the ideals of a field being trivial are a result of its division property and doesn't really invoke commutativity at all.

This result is a little more useful I think because it tells us what any given homomorphism will do when the domain is a division ring - that is, it forces injectivity.

I'm still a wiki-n00b as far as moving stuff around and merging and all that jazz is concerned. Would you be able to shift this to a page replacing "Field" with "Division Ring" and "homorphism" with "homomorphism"? (lol typo ftw) --Grambottle 16:24, 20 December 2008 (UTC)

The Division Ring result is exactly the same, it likewise "forces injectivity" - isomorphism is defined as a bijective homomorphism and as such is injective by definition. So, as I say, it's effectively equivalent to your version for fields.

One thing to do would be to link it to the Division Ring one and invoke the result about a mapping onto its image is a surjection. I'll get to it in a bit. --Matt Westwood 18:53, 20 December 2008 (UTC)