Remarks on the commutativity of rings

Abstract

Introduction. A celebrated theorem of N. Jacobson [7] asserts that if (1) Xn(z) =x for every x in a ring R, where n(x) is an integer greater than one, then R is commutative. In a recent paper [2], I. N. Herstein has shown that it is enough to require that (1) holds for those x in R which are commutators: x= [y, z] =yz-zy of two elements of R. The purpose of this note is to show that if R has no nonzero nilpotent ideals, we may restrict x in (1) to iterated commutators of any fixed degree. We also obtain a weaker result for arbitrary rings. An important tool in the proof of our results is a lemma which generalizes a result of Kaplansky [4] to the effect that the only elements of a primitive ring which commute with all commutators are in the center. This tool is also useful in extending and complementing some results of Divinsky [1] on commuting isomorphisms of simple rings. These extensions include some recent results of Posner [9] on derivations in prime rings. As a final remark, we indicate an exceedingly brief proof of Herstein's result [3 ] on Jordan derivations in prime rings. Although the proof of our first result follows that of Herstein very closely, we present a self-contained account for the convenience of the reader.