The Structure of Rings

Abstract

In the first part of this chapter a general structure theory for rings is presented. Although the concepts and techniques introduced have widespread application, complete structure theorems are available only for certain classes of rings. The basic method for determining such a class of rings might be described intuitively as follows. One singles out an “undesirable” property P that satisfies certain conditions, in particular, that every ring has an ideal which is maximal with respect to having property P. This ideal is called the P-radical of the ring. One then attempts to find structure theorems for the class of rings with zero P-radical. Frequently one must include additional hypotheses (such as appropriate chain conditions) in order to obtain really strong structure theorems. These ideas are discussed in full detail in the introductions to Sections 1 and 2 below. The reader would do well to read both these discussions before beginning serious study of the chapter.