Some Commutativity Conditions for Rings

Abstract

During the last 55 years there have been many results concerning conditions that force a ring to be commutative. These results were stimulated by Jacobson’s famous result and were extensively devel- oped by Herstein. This paper will survey the area by organizing the results according to whether they come from variations on Herstein’s conditions, depend on general polynomial conditions, depend on the presence of a derivation, or whether a ring has special properties that make commutativity more easily accessible. Finally, the most recent conditions concern product sets and lead to results in a new area of inquiry.