On the n—nilpotency

Abstract

In this paper we prove that, for every integer n ≥ 1, d ≥ m ≥ 2, the sum of m n- nilpotent ideals of an algebra may not be d- nilpotent. This leads to a similar result in group theory; the product of m n- nilpotent normal subgroups may not be d- nilpotent. We apply these results to solve Rozhkov’s question by constructing some n- finite, infinite p- groups generated by n + 1 conjugates.