Czechoslovak Journal of Mathematics, 52 (127) (2002), 545-552

A Family of Noetherian Rings
with their Finite Length Modules Under Control

Dedicated to Helmut Lenzing on the occasion of his 60th birthday

Abstract.
We investigate the category  mod R  where  R  is the  k -tensor product of an elementary  k -algebra A and  a V-ring  S  containing k  in its center.  Recall that a (right) V-ring  S  is characterized by the property that every simple (right) S -module Ej  is injective.  We show that the tensor product  Pj  of   A  and  Ej  forms a quasi-progenerator, and hence, by a result of K. Fuller induces a category equivalence  between  mod R  and the product of the categories  mod End Pj .  As a consequence we can:
 

1. Construct for each elementary  k -algebra  A  over a finite field  k  a nonartinian ring  R such that the categories  of finite length modules  mod A  and  mod R  are equivalent.
2. Find twisted versions  R  of algebras of wild representation type such that  R  itself has finite or tame representation type.
3. Describe for such rings  R  the minimal almost split morphisms in  mod R  and observe that almost all of these maps are not almost split in the category of all modules  Mod R .

Download  family.dvi  or family.ps .

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Last modified:  November 19, 2002, by Markus Schmidmeier