Compactness of the structure space of a ring

Abstract

Jacobson [I ] has shown that a topology may be defined on the set SA of primitive ideals of any nonradical ring A. With this topology SA is called the structure space of A. The topology is given by defining closure: if T = { p I is a set of primitive ideals then T is the set of primitive ideals which contain nf {p IpE T}. One of Jacobson's results is that if A has a unit then SA is compact. By working with open sets rather than closure we obtain a simpler proof of this fact and also a new sufficient condition for compactness: every 2-sided ideal of A is finitely generated. Let p, q, *, be points of SA (primitive ideals). For each xEA write (x) for the principal (2 sided) ideal generated by x, and let Ux-= {I P p (X) I