Krull-Schmidt fails for Artinian modules

Facchini, Alberto; Herbera, Dolors; Levy, Lawrence S.; Vámos, Peter

doi:10.1090/S0002-9939-1995-1277109-4

Krull-Schmidt fails for Artinian modules
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by Alberto Facchini, Dolors Herbera, Lawrence S. Levy and Peter Vámos PDF

Proc. Amer. Math. Soc. 123 (1995), 3587-3592 Request permission

Abstract:

We prove that the Krull-Schmidt theorem fails for artinian modules. This answers a question asked by Krull in 1932. In fact we show that if S is a module-finite algebra over a semilocal noetherian commutative ring, then every nonunique decomposition of every noetherian S-module leads to an analogous nonunique decomposition of an artinian module over a related non-noetherian ring. The key to this is that any such S is the endomorphism ring of some artinian module.

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Matrizen, Moduln und verallgemeinerte Abelsche Gruppen im Bereich der ganzen algebraischen Zahlen

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