Lie ideals in group rings

Abstract

Let U be a Lie ideal of an associative unitary ring R. It is proved that if U is Lie nilpotent of class n⩾3, then y 3( U) R is a nilpotent ideal of index at most 2 n−4 and every element of y 2( U) is nilpotent for a proper Lie ideal U(i.e. U≠ R) provided 2 is invertible in R. If R is a Lie nilpotent associative algebra of characteristic p>0, then the Lie commutators {[ x, y]= xy− y| x, yϵR} generate a nil ideal of bounded exponent. Further, if R= K[ G], the group algebra of a group G over a field K of characteristic p⩾0, such that y n ( L( K[ G]))=0, then for p=0 and for p⩾ n, G is an abelian group and for p< n, G is a nilpotent group of class at most c, where c is the least positive integer not less than 2( n−1) (p−1) if p>3 and not less than 1+√ n−4 if p=3. Some results of independent interest on Lie lower central chains are also obtained.