A Krull–Schmidt Theorem for Noetherian Modules,

We prove a version of the Krull–Schmidt theorem which applies to Noetherian modules. As a corollary we get the following cancellation rule: If A, B, C are nonzero Noetherian modules such that either A C B C or An Bn for some n , then there are modules A′ ≤ A and B′ ≤ B such that A′ B′ and len A′ = len A = len B′ = len B. Here the ordinal valued length, len A, of a module A is as defined in G. Brookfield [Comm. Algebra30 (2002), 3177–3204] and T. H. Gulliksen [J. Pure Appl. Algebra3 (1973), 159–170]. In particular, A, B, A′, and B′ have the same Krull dimension, and A/A′ and B/B′ have strictly smaller Krull dimensions than A and B.