Ziegler spectra of serial rings with krull dimension

Abstract

The (right) Ziegler spectrum of a ring R is a topological space the points of which are the isomorphism types of the indecomposable pure-injective (right) K-modules and is denoted by Z g R. The CB-rank of the Ziegler spectrum of a ring R is a measure for the complexity of the cat­egory of K-modules. The m-dimension of a module is a measure for the complexity of the lattice of its pp-definable subgroups. Using the clas­sification of the isomorphism types of the indecomposable pure-injective modules over a serial ring obtained by Eklof, Herzog and Puninski, we show that if R is a serial ring with Krull dimension, then Z g R has CB-rank and for every point in ZgR, its CB-rank is equal to its m-dimension. We also show that a serial Krull-Schmidt ring R has Krull dimension iff the largest theory of (right) R-modules has m-dimension.