International Journal of Mathematics and Mathematical Sciences
Volume 25 (2001), Issue 5, Pages 299-304
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 Adil Yaqub. 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.
A well-known theorem of Jacobson (1964, page 217) asserts that a
ring with the property that, for each in , there exists
an integer such that is necessarily
commutative. This theorem is generalized to the case of a weakly
periodic ring with a sufficient number of potent extended
commutators. A ring is called weakly periodic if every in can be written in the form ,
where is nilpotent and is potent in the sense
some integer . It is shown that a weakly periodic ring in which certain extended commutators are potent must have a nil
commutator ideal and, moreover, the set of nilpotents forms an ideal which, in fact, coincides with the Jacobson radical of .