COMMUTATIVITY OF RINGS WITH CONSTRAINTS ON NILPOTENTS AND NONNILPOTENTS

ABSTRACT. Let R be a ring (not necessarily with identity), N the set of nilpotents, and n> a fixed integer. Suppose that (i) N is commutative; (ii) If x N and n ny N, then x y xy (iii) For a N and b R, if n![a,b] 0, then [a,b] 0, where [a,b] ab ba denotes the commutator. Then R is commutative. This theorem generalizes the "xn--x " theorem of Jacobson. It is also shown that above theorem need not be true if any of the hypotheses is deleted, or if "n! " in (iii) is replaced by "n"