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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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


Authors: Birge Zimmermann-Huisgen and Wolfgang Zimmermann
Journal: Trans. Amer. Math. Soc. 320 (1990), 695-711
MSC: Primary 16A64; Secondary 16A53, 16A65, 16A90
DOI: https://doi.org/10.1090/S0002-9947-1990-0965304-0
MathSciNet review: 965304
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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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DOI: https://doi.org/10.1090/S0002-9947-1990-0965304-0
Article copyright: © Copyright 1990 American Mathematical Society