Let be a fixed field. Consider associative -algebra and its category of right modules .
Recall that a monomorphism in -modules is pure if after tensoring with any left -module gives a mono . A module in is pure-injective if every pure mono splits. This is clearly a weaker property than being an injective object.
Given an associative algebra , its Ziegler spectrum is the topological space whose points are the isomorphism classes of indecomposable pure-injective -modules and the topology is defined in terms of pp-formulas (or finite matrices) over . Here pp stands for “positive primitive in the usual language for -modules”
The importance of Ziegler spectrum is in the
Ziegler’s theorem. There is a correspondence between the definable classes in and closed subsets of .
There are applications to the spectra of theories of modules.