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International Journal of Mathematics and Mathematical Sciences
Volume 25, Issue 5, Pages 299-304
http://dx.doi.org/10.1155/S0161171201005051

Structure of weakly periodic rings with potent extended commutators

Department of Mathematics, University of California, Santa Barbara 93106, CA, USA

Received 1 July 1999; Revised 3 May 2000

Copyright © 2001 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A well-known theorem of Jacobson (1964, page 217) asserts that a ring R with the property that, for each x in R, there exists an integer n(x)>1 such that xn(x)=x is necessarily commutative. This theorem is generalized to the case of a weakly periodic ring R with a “sufficient” number of potent extended commutators. A ring R is called weakly periodic if every x in R can be written in the form x=a+b, where a is nilpotent and b is “potent” in the sense that bn(b)=b for some integer n(b)>1. It is shown that a weakly periodic ring R in which certain extended commutators are potent must have a nil commutator ideal and, moreover, the set N of nilpotents forms an ideal which, in fact, coincides with the Jacobson radical of R.