ON CENTER-LIKE ELEMENTS IN RINGS

ABSTRACT. In a paper with a similar title Herstein has considered the structure of inprime rings which contain an element a which satisfies either [a x 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]nwe consider are (i) [[a,xl],X2].,Xm 0 or is in the center of R for all Xl,X2.,x in R and (2) [a,[xl,[X2,..,[Xm_I x]]]n 0 for all Xl,X2 xm m m in R. We obtain results similar to those obtained by Herstein; however, in some cases we must strengthen the hypotheses n k Also we consider a third type (3) (axm x a) 0 for all x in R and extend results of Herstein and Giambruno. KEY WORDS AND PHRASES. Prime and semiprime rings, primitive and semiprimitive rings, rings with involution, commutativity theorems. 1980 MATHEMATICS SUBJECT CLASSIFICATION CODE. 16A12, 16A20, 16A28, 16A70 i. INTRODUCTION. The definition of the center Z of a ring R has recently been generalized i