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A family of Noetherian rings with their finite length modules under control. (English) Zbl 1014.16014

Summary: We investigate the category \(\text{mod }\Lambda \) 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 \(\text{mod }\Lambda\simeq\coprod_j\text{mod End }P_j\). As a consequence we can (1) construct for each elementary \(k\)-algebra \(A\) over a finite field \(k\) a non-Artinian Noetherian ring \(\Lambda\) such that \(\text{mod }A\simeq\text{mod }\Lambda\), (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 \(\text{mod }\Lambda\) and observe that almost all of these maps are not almost split in \(\text{Mod }\Lambda\).

MSC:

16G10 Representations of associative Artinian rings
16G60 Representation type (finite, tame, wild, etc.) of associative algebras
16G70 Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
16D90 Module categories in associative algebras
16P40 Noetherian rings and modules (associative rings and algebras)
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References:

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