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

Abstract

We investigate the category modΛ of finite length modules over the ring Λ=A ⊗ k Σ, where Σ 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 ⊗ E j . By a result of K. Fuller, P j induces a category equivalence from which we deduce that modΛ ≃ ∐j mod EndP j . As a consequence we can (1) construct for each elementary k-algebra A over a finite field k a nonartinian noetherian ring Λ such that modA ≃ modΛ (2) find twisted versions Λ of algebras of wild representation type such that Λ itself is of finite or tame representation type (in mod) (3) describe for certain rings Λ the minimal almost split morphisms in modΛ and observe that almost all of these maps are not almost split in ModΛ.