Krull–schmidt fails for artinian modules

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

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26 Citations (Scopus)


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. © 1995 American Mathematical Society.
Original languageEnglish
Pages (from-to)3587-3592
JournalProceedings of the American Mathematical Society
Issue number12
Publication statusPublished - 1 Jan 1995


  • Capacities
  • Direct sum decomposition
  • Endomorphism ring
  • Krull-Schmidt


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