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  "body": "\n## Summary\n\nAn indecomposable module is a nonzero [[module|0NZ8JA25WK]] that cannot be written as a direct sum of two nonzero submodules.\nEvery [[simple module|G00C22AEDH]] is indecomposable, but an indecomposable module may have many submodules.\nBy the Krull–Schmidt theorem, a module of finite length is a direct sum of indecomposable modules in an essentially unique way.\nSo classifying modules of finite length over a ring comes down to classifying its indecomposable modules.\n\n## Explanations\n\n### L1 human\n\nAn indecomposable module cannot be split into two independent nonzero parts.\nIt may still have smaller pieces inside, but none of them can be split off cleanly.\nThe group $\\Z/4\\Z$ is an example: it contains $\\set{0, 2}$, but no subgroup is a complement to it.\nThe group $\\Z/6\\Z$ is not indecomposable, since it is $\\Z/2\\Z \\oplus \\Z/3\\Z$.\n",
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