International Journal of Mathematics and Mathematical Sciences
Volume 8 (1985), Issue 1, Pages 145-149
On center-like elements in rings
University of Cincinnati, Cincinnati 45221, Ohio, USA
Received 4 June 1982; Revised 1 September 1982
Copyright © 1985 Joe W. Fisher and Mohamed H. Fahmy. 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.
In a paper with a similar title Herstein has considered the structure of prime rings which contain an element a which satisfies either or is in the center of for each in . We consider the structure of rings which contain an element a which satisfies powers of certain higher commutators. The two types which we consider are (1) or is in the center of for all in and (2) for all in . We obtain results similar to those obtained by Herstein; however, in some cases we must strengthen the hypotheses.
Also we consider a third type (3) for all in and extend results of Herstein and Giambruno.