Structure of rings satisfying certain identities on commutators

Abstract:

Suppose R is an associative ring with center Z, and suppose J is the Jacobson radial of R. Suppose that, for all x, y in R, there exist ${z_{x,y}} \in Z$ and an integer $n(x,y) > 1$ such that \begin{equation}\tag {$A$} xy - yx = {(xy - yx)^{n(x,y)}}{z_{x,y}}.\end{equation} Then $R/J$ is a subdirect sum of division rings satisfying: ${(xy - yx)^{n(x,y) - 1}}$ is in the center. Additional results on the additive and multiplicative commutators which are in the center of a division ring satisfying (A) are also obtained. Next, suppose D is a division ring of characteristic not 2 and with the property that, for some x, y in D, there exists a positive integer n such that ${(xy - yx)^n}$ is in the center, and suppose that the smallest such n is even, then D contains a subdivision ring isomorphic to the “generalized” quaternions (and conversely). Finally, it is proved that an arbitrary ring R with the property that for all x, y in R, there exists ${z_{x,y}}$ in Z such that $xy - yx = {(xy - yx)^2}{z_{x,y}}$ is necessarily commutative, and that the exponent 2 cannot, in general, be replaced by 3.