On the sparsity of representations of rings of pure global dimension zero

Zimmermann-Huisgen, Birge; Zimmermann, Wolfgang

doi:10.1090/S0002-9947-1990-0965304-0

On the sparsity of representations of rings of pure global dimension zero
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by Birge Zimmermann-Huisgen and Wolfgang Zimmermann PDF

Trans. Amer. Math. Soc. 320 (1990), 695-711 Request permission

Abstract:

It is shown that the rings $R$ all of whose left modules are direct sums of finitely generated modules satisfy the following finiteness condition: For each positive integer $n$ there are only finitely many isomorphism types of (a) indecomposable left $R$-modules of length $n$; (b) finitely presented indecomposable right $R$-modules of length $n$; (c) indecomposable right $R$-modules having minimal projective resolutions with $n$ relations. Moreover, our techniques yield a very elementary proof for the fact that the presence of the above decomposability hypothesis for both left and right $R$-modules entails finite representation type.

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