On center-like elements in rings

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 , x 1 ] , x 2 ] , … x m ] n = 0 or is in the center of R for all x 1 , x 2 , … , x m in R and (2) [ a , [ x 1 , [ x 2 , … , [ x m − 1 , x m ] … ] ] ] n for all x 1 , x 2 , … , x m 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) ( a x m − x n a ) k = 0 for all x in R and extend results of Herstein and Giambruno.