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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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 $ {z_{x,y}} \in Z$ and an integer $ n(x,y) > 1$ such that

$\displaystyle xy - yx = {(xy - yx)^{n(x,y)}}{z_{x,y}}.$ ($ A$)

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.

References [Enhancements On Off] (What's this?)

  • [1] I. N. Herstein, A condition for the commutativity of rings, Canad. J. Math. 9 (1957), 583–586. MR 0091941,
  • [2] Nathan Jacobson, Structure of rings, American Mathematical Society Colloquium Publications, Vol. 37. Revised edition, American Mathematical Society, Providence, R.I., 1964. MR 0222106
  • [3] Neal H. McCoy, The theory of rings, The Macmillan Co., New York; Collier-Macmillan Ltd., London, 1964. MR 0188241
  • [4] O. T. O'Meara, Introduction to quadratic forms, Die Grundlehren der math. Wissenschaften, Band 117, Academic Press, New York; Springer-Verlag, Berlin, 1963. MR 27 #2485.

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