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  "body": "\n## Summary\n\nA simple module is a nonzero [[module|0NZ8JA25WK]] whose only [[submodules|05WDSN597C]] are zero and the module itself.\nSimple modules are the building blocks of modules of finite length: the factors of a [[composition series|0XV2A49X6Q]] are simple.\nOver a group algebra, simple modules are the same as [[irreducible representations|FX3NEFYFFN]].\nEvery simple left module over a ring $R$ is isomorphic to $R/\\mathfrak{m}$ for a maximal left ideal $\\mathfrak{m}$.\n\n## Explanations\n\n### L1 human\n\nA simple module cannot be broken into a smaller nonzero piece inside it.\nIt plays the role that prime numbers play for integers, or that simple groups play for finite groups.\nFor the integers, the simple modules are the groups $\\Z/p\\Z$ with $p$ prime.\nSuch a group has $p$ elements, and a subgroup has an order dividing $p$, so it is either zero or everything.\n",
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