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.
A finite-dimensional module is with pairwise nonisomorphic indecomposables and uniquely determined multiplicities. The Fitting lemma makes their endomorphism rings local. The multiplicity spaces provide the general linear factors in the Levi decomposition of a quiver automorphism group.
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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.