Strong preinjective partitions and representation type of Artinian rings

Zimmermann-Huisgen, Birge

doi:10.1090/S0002-9939-1990-1007520-3

Strong preinjective partitions and representation type of Artinian rings
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by Birge Zimmermann-Huisgen

Proc. Amer. Math. Soc. 109 (1990), 309-322

DOI: https://doi.org/10.1090/S0002-9939-1990-1007520-3

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Abstract:

It is shown that for every ring of left pure global dimension zero (i.e., for every ring all of whose left modules are direct sums of finitely generated modules), the finitely generated left modules can be grouped to a unique "strong preinjective partition" while the finitely presented right modules possess a "strong preprojective partition"; these strong partitions are upgraded versions of the partitions introduced by Auslander and Smaløfor Artin algebras. One direct consequence is that a ring of left pure global dimension zero has finite representation type if and only if there exist sufficiently many almost split maps among its finitely generated left modules. This provides a very elementary proof for Auslander’s theorem saying that for Artin algebras vanishing of the left pure global dimension is equivalent to finiteness of the representation type.

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