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)

Okay, moved, make sure the name is right. I'm not sure if non-sysops can move a page, look on the bar with the edit button for the word move. And as for merging, you have to do it manually, just copy and paste what you want from one page to the other. --Cynic-(talk) 02:04, 21 December 2008 (UTC)

Sorry but I still think this page is superfluous. Both this and Epimorphism from Division Ring to Ring effectively say exactly the same thing. --Matt Westwood 06:39, 21 December 2008 (UTC)

It may be superfluous, but its much better to have some pages we don't need than to need some pages we don't have. It's not like were wasting paper, as you've said before. And besides, it might not be obvious to everyone that the proofs are basically identical. --Cynic-(talk) 19:38, 21 December 2008 (UTC)

Yeah okay, fair enough ... we have to be careful we don't get too fragmented. If two parallel streams of proofs start to develop with no reference between them and the only material difference between them is their names, then we may need to refactor ... --Matt Westwood 22:11, 21 December 2008 (UTC)