International Journal of Mathematics and Mathematical Sciences
Volume 8 (1985), Issue 1, Pages 145-149
doi:10.1155/S0161171285000138
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.
Abstract
In a paper with a similar title Herstein has considered the structure of prime rings which contain an element a which satisfies either [a,x]n=0 or is in the center of R for each x in R. 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) […[[a,x1],x2],…xm]n=0 or is in the center of R for all x1,x2,…,xm in R and (2) [a,[x1,[x2,…,[xm−1,xm]…]]]n for all x1,x2,…,xm in R. 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) (axm−xna)k=0 for all x in R and extend results of Herstein and Giambruno.