When nilpotent elements generate nilpotent ideals

Abstract

We study the natural class of rings where each nilpotent element generates a nilpotent ideal, calling them the strongly[Formula: see text]-primal rings. We derive many basic properties of these rings, analyze their behavior under standard ring constructions and extensions, and taxonomize their relationship to other natural generalizations of commutativity. A slightly stronger condition is to assume that any nilpotent element generates a nilpotent ideal of the same index of nilpotence. We find that the difference between these two properties explains divergent behaviors in direct products, polynomial rings, Morita contexts, and other constructions.