Talk:Ore condition

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)

Some necessary clarifications
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" $$R = k$$ 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)


 * 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)