Rings with nil and power centralK-th commutators

We define thek-th commutator forx, y in a ringR inductively as follows: x,y]₁=x,y]=xy−yx and x,y] _k =x,y] _{k−1, y} ]. Assume thatR is a ring without nonzero nil onesided ideals. The following are shown: (1) If x,y] _k is nilpotent for allx,y∈R, thenR must be commutative. (2) If x,y] _k is power central for allx,y∈R, thenR must satisfy the standard polynomial of degree 4. 1980 Mathematics Subject Classification (1985 Revision). Primary 16A70, Secondary 16A12.