Rings without finiteness assumption

Abstract

For general rings there is naturally not as much structure theory as in the Artinian or Noetherian case. It is true that some of the same ethods can be used, e.g. the radical can be defined, semiprimitive rings can be xpressed as subdirect products of primitive rings etc., but these methods are less precise and they do not lead to a complete classification. For primitive rings a structure theorem can be proved using a general version of the density theorem; this is presented in Section 8.1 and applied in Section 8.2, while Section 8.3 deals with semiprimitive rings. So far we have taken the existence of a unit element for granted, but some work has been done on `rings without one' and we cast a brief glance at it in ection 8.4; we shall examine the case of simple rings and also see when the existence of a `one' follows from other assumptions. In Section 8.5 we study semiprime rings, and in Section 8.6 we present an analogue of Goldie's theorem for PI-rings. The final section, Section 8.7, takes a brief look at a natural generalization of principal ideal domains: free ideal rings.