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

DOI:
https://doi.org/10.1090/S0002-9939-1972-0291219-4

MathSciNet review:
0291219

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Abstract | References | Similar Articles | Additional Information

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.

**[1]**I. N. Herstein,*A condition for the commutativity of rings*, Canad. J. Math.**9**(1957), 583–586. MR**0091941**, https://doi.org/10.4153/CJM-1957-066-0**[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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DOI:
https://doi.org/10.1090/S0002-9939-1972-0291219-4

Article copyright:
© Copyright 1972
American Mathematical Society