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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Krull-Schmidt fails for Artinian modules


Authors: Alberto Facchini, Dolors Herbera, Lawrence S. Levy and Peter Vámos
Journal: Proc. Amer. Math. Soc. 123 (1995), 3587-3592
MSC: Primary 16P20; Secondary 16D70
MathSciNet review: 1277109
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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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1995-1277109-4
PII: S 0002-9939(1995)1277109-4
Keywords: Krull-Schmidt, artinian module, direct sum decomposition, endomorphism ring
Article copyright: © Copyright 1995 American Mathematical Society