# Are the elements of a division algebra which commute with all commutators in the center of the algebra?

I asked this quetion five days ago at http://math.stackexchange.com/questions/406669/are-the-elements-of-a-division-algebra-which-commute-with-all-commutators-in-the Some good people have given good comments there.

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This is a lemma in Ancohea's paper jstor.org/discover/10.2307/…. I can not understand the proof. –  yanyu Jun 3 '13 at 4:43
Hmm, the proof of the lemma in that paper seems to suggest Ancohea assumes the division algebra is finite-dimensional over the center, but this is never clearly stated earlier in the paper. How about the following 1-sentence proof: the commutators over $k$ span the space of commutators over $\overline{k}$, so it suffices to treat the case of a matrix algebra, which you can treat by bare hands. QED –  user30180 Jun 3 '13 at 6:22