Primitive Rings and the Density Theorem

Abstract

We saw in Theorem 1.15 that simple artinian rings are precisely those artinian rings which have a faithful simple module. It is useful to drop the finiteness condition and to study those rings which have a faithful simple module but are not necessarily artinian. Such a ring is called a primitive ring. Primitive rings, a generalization of simple rings, play a role analogous to that of simple rings in that they may be viewed as the basic building blocks of other rings, though in an extended, infinite dimensional context. This perhaps justifies the name primitive. The theory of primitive rings can be developed along lines parallel to that of simple rings. The two theories intertwine, and in fact some authors choose to study simple rings from the point of view of primitive rings. This chapter explores such an approach.