Talk:Ore condition

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There is PlanetMath content for this, but spread over several pages. I think it should be an integrated discussion, here. Charles Matthews 11:29, 7 November 2005 (UTC)Reply

I'm afraid the article at the moment contains a couple of mistakes. I checked out both the Planet Math first reference, and P. M. Cohns book Skew fields

1. The right and left Ore conditions are reversed, compared to the use in either of the sources (which also happens to be the way I remember it). In other words, "left" and "right" should be interchanged in the article.
2. Either Ore condition is equivalent to the existence of a very special kind of embedding into a division ring, the (right or left) classical ring of quotient. However, a general subring R of a division ring D may fail to fulfil one or both of the Ore conditions. As an example of the latter (due to P. M. Cohn, 1971), consider a "non commutative polynomial ring" ${\displaystyle R=k<x,y>}$ in two variables over a (commutative) field; in other words, R is the monoid algebra over k with respect to the free monoid on two generators x and y. It is not very hard to see that R does not fulfil either Ore condition; but by Cohn's result, it is indeed isomorphic to a subring of a certain division ring.

I'll see if I can add this example, and an example of a ring fulfilling just one of the Ore conditions, to the article, sometime in the future; right now, I'll just correct the errors in the most simple manner.--JoergenB (talk) 19:18, 16 December 2007 (UTC)Reply

I tweaked stuff a bit with the aim of highlighting the importance of the special embedding, but I forgot to explain in the edit notes. Rschwieb (talk) 15:18, 16 May 2011 (UTC)Reply

EDIT: It turns out that the article did quote correctly from (Cohn 1995, Cor 4.5.9) harv error: no target: CITEREFCohn1995 (help).

This is what Cor 4.5.9 says:

> For any semifir (and in particular, any fir) R, the localization ${\displaystyle R_{\Phi }}$ at the set of all full matrices is a universal field of fractions for R, with a fully inverting homomorphism ${\displaystyle R\rightarrow R_{\Phi }}$.

In the Preface, which is currently accessible here https://www.maths.ed.ac.uk/~v1ranick/papers/cohnfr.pdf, Cohn makes clear that by "field" he actually means "skew field". He also seems to imply that the map above is an injective map. I'll need to read the source more carefully to be sure. Svennik (talk) 00:20, 29 March 2025 (UTC)Reply