Definable subcategories and the ziegler spectrum

Abstract

In (ZIEGLER, 1984) Ziegler associated a topological space to the category of modules over any associative ring with unit. This space, now known as the Ziegler Spectrum, has as points the isomorphism classes of pure-injective indecomposable modules. This topological space is able to give a better understanding to the category of modules. The main objective of this text is to give some necessary definitions to understand the Ziegler spectrum and proof some important results about it. The focus of the text are definable subcategories of Mod-R, defining the Ziegler spectrum, proof some results related to it and give the example of the Ziegler Spectrum for discrete valuation rings.