A family of noetherian rings with their finite length modules under control

Author: Schmidmeier M.¹

Source: Czechoslovak Mathematical Journal, Volume 52, Number 3, September 2002 , pp. 545-552(8)

Publisher: Springer

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Abstract:

We investigate the category \rm mod\Lambda of finite length modules over the ring \Lambda=A\otimes_{k}\Sigma, where \Sigma is a V-ring, i.e. a ring for which every simple module is injective, k a subfield of its centre and A an elementary k-algebra. Each simple module E_j gives rise to a quasiprogenerator P_j=A\otimes E_{j}. By a result of K. Fuller, P_j induces a category equivalence from which we deduce that {\rm mod}\Lambda \simeq {\coprod}_j\ {\rm mod \ End}P_j. As a consequence we can

(1) construct for each elementary k-algebra A over a finite field k a nonartinian noetherian ring \Lambda such that {\rm mod}A \simeq {\rm mod}\Lambda,

(2) find twisted versions \Lambda of algebras of wild representation type such that \Lambda itself is of finite or tame representation type (in mod),

(3) describe for certain rings \Lambda the minimal almost split morphisms in \rm mod\Lambda and observe that almost all of these maps are not almost split in \rm Mod\Lambda.

Keywords: V-ring; progenerator; almost split morphisms

Language: English

Document Type: Research article

Affiliations: 1: Dept. of Math. Sciences, Florida Atlantic University, Boca Raton, Florida 33431-0991, U.S.A., mschmidm@fau.edu

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