Bounds for indices of nilpotency of finitely generated nil rings of finite index

An improved bound is given for the index of nil-potency of a finitely generated nil ring of index n in terms of the index of nilpotency of the ideal generated by T^mwhere m = [n/2] and T is a m-subset of the set of generators. If m = 3 it is proved that T¹⁰is contained in an ideal generated by twenty-seven cubes and this is applied to get bounds for the index of nilpotency of a finitely generated nil ring of index 6 or 7, bounds which are less than one hundredth of the bounds we obtained in a previous paper. 1980 AMS subject classification. Primary 16A22; Secondary 16438.