Introduction

Summary

The Ziegler spectrum, ZgR, of a ring R is a topological space. It is defined in terms of the category of R-modules and, although a Ziegler spectrum can be assigned to much more general categories, let us stay with rings and modules at the beginning. The points of ZgR are certain modules, more precisely they are the isomorphism types of indecomposable pure-injective (also called algebraically compact) right R-modules. Any injective module is pure-injective but usually there are more, indeed a ring is von Neumann regular exactly if there are no other pure-injective modules (2.3.22). If R is an algebra over a field k, then any module which is finitedimensional as a k-vector space is pure-injective (4.2.6). Every finite module is pure-injective (4.2.6). Another example is the ring of p-adic integers, regarded as a module over any ring between ℤ and itself (4.2.8). The pure-injective modules mentioned so far are either “small” or, although large in some sense, have some kind of completeness property. There is something of a general point there but, as it stands, it is too vague: not all “small” modules are pure-injective. For example, the finite-length modules over the first Weyl algebra, A₁(k), over a field k of characteristic zero are not pure-injective (8.2.35).