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  "body": "\n## Statement\n\nLet $R$ be a ring and let $M$ be a nonzero left [[$R$-module|09VDS2W1E8]] of [[finite length|V6JBKT168T]].\n\n1. $M$ is a direct sum $M = M_1 \\oplus \\cdots \\oplus M_m$ of finitely many [[indecomposable|E2P29GCD9T]] submodules.\n2. If also $M = N_1 \\oplus \\cdots \\oplus N_n$ with each $N_j$ indecomposable, then $m = n$ and there is a permutation $\\sigma$ of $\\set{1, \\dots, m}$ with $M_i \\cong N_{\\sigma(i)}$ for all $i$.\n\n## Remarks\n\nExistence follows by induction on the length: a module that is not indecomposable splits into two summands of smaller length.\nFor uniqueness, [[Fitting's lemma|F52B2JMEHY]] shows that the endomorphism ring of each indecomposable summand is local.\nAn exchange argument then matches $M_1$ with some $N_j$ and replaces one by the other, and induction on $m$ finishes.\nExample: a finite abelian group is a $\\Z$-module of finite length, and its decomposition into cyclic groups of prime power order is unique up to order and isomorphism.\nUniqueness can fail without finite length: in the ring of integers $\\mathcal{O}$ of a number field with nontrivial class group, a non-principal ideal $I$ satisfies $I \\oplus I^{-1} \\cong \\mathcal{O} \\oplus \\mathcal{O}$, though $I \\not\\cong \\mathcal{O}$.\n\n## Explanations\n\n### L1 human\n\nA module of finite length breaks into pieces that cannot be broken further.\nThe theorem says that the list of pieces is determined by the module, up to reordering and isomorphism, just as the prime factors of an integer are.\n\n### L2 human\n\nThe theorem reduces the study of modules of finite length to that of indecomposable modules.\nFor finite-dimensional algebras over a field, every finite-dimensional module has finite length, so the theorem applies; this is the base of representation theory of quivers and algebras.\nThe same proof works in any additive category in which idempotents split and endomorphism rings of indecomposable objects are local.\n",
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