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  "body": "\n## Statement\n\nLet $R$ be a ring, let $M$ be a left [[$R$-module|09VDS2W1E8]] of [[finite length|V6JBKT168T]], and let $f \\in \\End_R(M)$ be an element of its [[endomorphism ring|AWSBZWWMG9]].\nThen there is an integer $n \\ge 1$ such that:\n\n1. $M = \\im f^n \\oplus \\ker f^n$, a direct sum of [[submodules|0RGZ10DFPG]];\n2. the restriction of $f$ to $\\im f^n$ is an automorphism of $\\im f^n$;\n3. the restriction of $f$ to $\\ker f^n$ is nilpotent.\n\nIn particular, if $M$ is [[indecomposable|E2P29GCD9T]], then every endomorphism of $M$ is either an automorphism or nilpotent.\n\n## Remarks\n\nA module of finite length is both Noetherian and Artinian.\nSo the chain $\\ker f \\subseteq \\ker f^2 \\subseteq \\cdots$ and the chain $\\im f \\supseteq \\im f^2 \\supseteq \\cdots$ both become constant, say from step $n$ on.\nIf $x \\in \\im f^n \\cap \\ker f^n$, then $x = f^n(y)$ with $f^{2n}(y) = 0$, so $y \\in \\ker f^{2n} = \\ker f^n$ and $x = 0$.\nFor $x \\in M$, pick $z$ with $f^n(x) = f^{2n}(z)$; then $x = f^n(z) + (x - f^n(z))$ splits $x$ into the two parts.\nFor an indecomposable $M$ the corollary shows that the non-invertible endomorphisms form an ideal, so $\\End_R(M)$ is a local ring.\nFinite length is needed: multiplication by $2$ on the indecomposable $\\Z$-module $\\Z$ is neither an automorphism nor nilpotent.\n\n## Explanations\n\n### L1 human\n\nRepeat a self-map of a module of finite length many times.\nEventually the module splits into two parts: on one part the map keeps shuffling elements bijectively, and on the other it pushes everything to zero.\nIf the module cannot be split at all, only one of these behaviours can occur.\n\n### L2 human\n\nFitting's lemma is the module version of the splitting of a linear map of a finite-dimensional space into an invertible part and a nilpotent part.\nIts main use is the corollary that the endomorphism ring of an indecomposable module of finite length is local.\nThis locality is exactly what the proof of the [[Krull–Schmidt theorem|94M0BPV1W4]] needs.\n",
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