Krull–Schmidt–Azumaya theorem

Krull–Schmidt–Azumaya theorem: Let $M$ be an $R$-module that has finite length (equivalently, $M$ admits a composition series ). Then:

1. $M$ decomposes as a finite direct sum of indecomposable submodules.
2. Any two decompositions of $M$ into finite direct sums of indecomposable modules are equivalent up to permutation and isomorphism of summands: if $M\cong \bigoplus_{i=1}^n M_i \cong \bigoplus_{j=1}^m N_j$ with all $M_i,N_j$ indecomposable, then $n=m$ and after reindexing $M_i\cong N_i$ for all $i$.

A common sufficient hypothesis for Krull–Schmidt is that $M$ is both Artinian and Noetherian (which implies finite length). The theorem underlies the uniqueness of indecomposable decompositions and contrasts with the stronger behavior of semisimple modules where summands can be taken simple.

Proof sketch (optional): Existence follows from descending chain conditions on direct summands, splitting off indecomposables inductively. Uniqueness uses that endomorphism rings of indecomposable finite-length modules are local, enabling a cancellation argument that matches summands between two decompositions.