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,,[xm1,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) (axmxna)k=0 for all x in R and extend results of Herstein and Giambruno.