Representations of the category of serial modules of finite Goldie dimension

We study the category SUsr of all serial right modules of finite Goldie dimension over a fixed ring R. This category has natural valuations m-dimU and e-dimU for every uniserial right R-module U, because the number m-dimU(A) (e-dimU(A)) of modules in the same monogeny (epigeny) class as U in any indecomposable direct-sum decomposition of an object A of SUsr is uniquely determined by A. These valuations m-dimU, e-dimU: V (SUsr) → N0 are not essential valuations in general, so that the quotient categories of SUsr corresponding to the valuations m-dimU, e-dimU are Krull-Schmidt categories but not IBN categories. With these valuations, we obtain a natural representation SUsr → <i∈I Ai that is an isomorphism reflecting and direct-summand reflecting additive functor. Here the categories Ai are quotient Krull-Schmidt categories of SUsr. If, instead of considering all the valuations m-dimU, e-dimU of SUsr that come from the monogeny classes and the epigeny classes of all uniserial right R-modules, we consider only those that are essential valuations, then we get a divisor theory of the monoid V (SUsr) and, correspondingly, a representation SUsr → <i∈J Ai in which the categories Ai, i ∈ J, are IBN categories. © de Gruyter 2008.