KRULL-SCHMIDT FAILS FOR ARTINIAN-MODULES

We prove that the Krull-Schmidt theorem fails for artinian modules. Th is 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 r ing, then every nonunique decomposition of every noetherian S-module l eads to an analogous nonunique decomposition of an artinian module ove r a related non-noetherian ring. The key to this is that any such S is the endomorphism ring of some artinian module.