Prime and Primitive Rings

Abstract

In commutative ring theory, three basic classes of rings are: reduced rings, integral domains, and fields. The defining conditions for these classes do not really make any use of commutativity, so by using exactly the same conditions on rings in general, we can define (and we have defined) the notions of reduced rings, domains, and division rings. However, a little careful thought will show that this is not the only way to generalize the former three classes. In fact, the defining conditions for these classes are conditions on elements of a ring. When we move from commutative rings to noncommutative rings, an alternative way of generalizing an “element-wise” condition should be to replace the role of elements by that of ideals. By making these changes judiciously in the basic definitions, we are led to the notions of semiprime rings, prime rings, and (left or right) primitive rings.