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

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.