Krull-Schmidt fails for Artinian modules

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. Krull asked, in [K '32, p. 38], whether the Krull-Schmidt theorem holds for artinian modules; that is, whether M1 ED ... ED Mm N ED ... ED Nn with each Mi and Ni an indecomposable artinian right module over some ring R, implies that m = n and, after a suitable renumbering of the summands, each Mi -Ni . Warfield showed [W '69, Proposition 5] that the answer is yes when the ring R is either right noetherian or commutative. He did this by showing that, over any ring, any artinian indecomposable module that is a union of modules of finite length has a local endomorphism ring. [Recall that direct sums of indecomposable modules with local endomorphism rings have unique directsum decompositions, even when the direct sum contains infinitely many terms.] Subsequently, Camps and Dicks showed that the endomorphism ring of any artinian module is semilocal (i.e. semisimple artinian modulo its Jacobson radical) [CD '93]. They concluded that artinian modules cancel from direct sums; that is, if M@ A _ M@ B with M artinian, then A _ B. In this note we answer Krull's question by showing that the Krull-Schmidt theorem fails for general artinian modules. The idea of the proof is that decompositions of modules correspond to decompositions of their endomorphism ring, in a natural way. This reduces the question to that of what kinds of Received by the editors October 25, 1993 and, in revised form, May 2, 1994. 1991 Mathematics Subject Classification. Primary 16P20, 16D70.