On division rings with algebraic commutators of bounded degree. (Q1434135)

Using results on generalized polynomial and rational identities, the authors prove results about algebras with algebraic commutators. One result is that if \(R\) is a prime ring with extended centroid \(C\), and if for a positive integer \(n\) and all \(x,y\in R\), \(xy-yx\) is algebraic over \(C\) of degree at most \(n\), then \([RC:C]\leq n^2\). For the other results, let \(D\) be a division ring of characteristic \(0\) with center \(Z\) and \(n\) a positive integer. Then \([D:Z]\leq n^2\) if the following elements are algebraic over \(Z\) of degree at most \(n\): all \(E(x)\) for \(x\in D\) and \(E\) a nonzero derivation of \(D\); all \(axa^{-1}x^{-1}\) for \(x\in D\) and \(a\) noncentral in \(D\); or all \(xyx^{-1}y^{-1}\) for \(x,y\in D\) (no characteristic assumption here).