The Krull-Schmidt theorem is a fundamental result in the theory of modules and abelian categories, particularly in the context of decomposition of modules. It provides conditions under which a module can be decomposed into a direct sum of indecomposable modules, and offers a uniqueness aspect to this decomposition.
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The Krull–Schmidt theorem says that a finite-length module decomposes as a finite direct sum of indecomposable modules, uniquely up to permutation and isomorphism of the summands.