On Rings Whose Associated Lie Rings Are Nilpotent

We call (i?) 1 the Lie ring associated with R, and denote it by 9Î. The question of how far the properties of SR determine those of R is of considerable interest, and has been studied extensively for the case when R is an algebra, but little is known of the situation in general. In an earlier paper the author investigated the effect of the nilpotency of 9î upon the structure of R if R contains a nilpotent ideal N such that R/N is commutative. In the present note we prove that, for an arbitrary ring R, the nilpotency of 9î implies that the commutators of R of the form x o y generate a nil-ideal, while the commutators of R of the form (x o y) o z generate a nilpotent ideal (cf. §3). If R is finitely generated, and 9Î is nilpotent then the ideal generated by the commutators x o y is also nilpotent (cf. §4).