Structure of rings satisfying certain identities on commutators
Authors: Mohan S. Putcha, Robert S. Wilson and Adil Yaqub
Journal: Proc. Amer. Math. Soc. 32 (1972), 57-62
MSC: Primary 16A48
MathSciNet review: 0291219
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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 and an integer such that
Then is a subdirect sum of division rings satisfying: 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 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 in Z such that is necessarily commutative, and that the exponent 2 cannot, in general, be replaced by 3.
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