Memoirs of the American Mathematical Society 2001; 125 pp; softcover Volume: 149 ISBN10: 0821826182 ISBN13: 9780821826188 List Price: US$57 Individual Members: US$34.20 Institutional Members: US$45.60 Order Code: MEMO/149/707
 These notes present an introduction into the spectrum of the category of modules over a ring. We discuss the general theory of pureinjective modules and concentrate on the isomorphism classes of indecomposable pureinjective modules which form the underlying set of this spectrum. The interplay between the spectrum and the category of finitely presented modules provides new insight into the geometrical and homological properties of the category of finitely presented modules. Various applications from representation theory of finite dimensional algebras are included. Readership Graduate students and research mathematicians interested in associative rings and algebras. Table of Contents  Introduction
 The functor category
 Definable subcategories
 Left approximations
 duality
 Ideals in the category of finitely presented modules
 Endofinite modules
 KrullGabriel dimension
 The infinite radical
 Functors between module categories
 Tame algebras
 Rings of definable scalars
 Reflective definable subcategories
 Sheaves
 Tame hereditary algebras
 Coherent rings
 Appendix A. Locally coherent Grothendieck categories
 Appendix B. Dimensions
 Appendix C. Finitely presented functors and ideals
 Bibliography
