The (right)Ziegler spectrumof a ringRis a topological space the points of which are the isomorphism types of the indecomposablepure-injective(right)K-modules and is denoted byZgR. TheCB-rankof the Ziegler spectrum of a ringRis a measure for the complexity of the cat­egory ofK-modules. Them-dimensionof a module is a measure for the complexity of the lattice of itspp-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, weshow that if R is a serialring with Krull dimension, thenZgRhas 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 ringRhas Krull dimension iff the largest theory of (right)R-modules hasm-dimension.