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  "body": "\n## Statement\n\nLet $R$ be a ring and let $M$ be a left [[$R$-module|09VDS2W1E8]].\nThe following are equivalent:\n\n1. $M$ is [[semisimple|N3MRHDKDGC]], that is, a direct sum of [[simple|G00C22AEDH]] submodules;\n2. $M$ is a sum of simple submodules;\n3. every [[submodule|0RGZ10DFPG]] $N$ of $M$ is a direct summand: there is a submodule $N'$ with $M = N \\oplus N'$.\n\n## Remarks\n\nThe step from 2 to 1, and to 3, uses [[Zorn's lemma|04RXTQA5ZA]].\nGiven $M = \\sum_{i \\in I} S_i$ and a submodule $N$, choose a set $J \\subseteq I$ maximal such that the sum $N + \\sum_{j \\in J} S_j$ is direct.\nEach $S_i$ meets this sum, and being simple it lies inside it, so the sum is all of $M$; taking $N = 0$ gives 1.\nFor 3 implies 2, one shows that every nonzero submodule contains a simple submodule, and that the sum of all simple submodules has a complement which must be zero.\nConsequences: submodules and quotient modules of semisimple modules are semisimple.\nExample: in a vector space every subspace has a complementary subspace, matching condition 3.\n\n## Explanations\n\n### L1 human\n\nThere are three ways to say that a module is built cleanly from simple pieces.\nIt is a direct sum of simple pieces; or it is merely covered by simple pieces, possibly overlapping; or every piece of it can be split off.\nThe theorem says these all mean the same thing.\n\n### L2 human\n\nCondition 3 is usually the easiest to check, and it is how Maschke's theorem proves semisimplicity of group representations.\nCondition 2 shows quickly that quotients of semisimple modules are semisimple.\nThe equivalence is the starting point of the theory of semisimple rings, where every module satisfies these conditions.\n",
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  "msc": [
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  "name": "Characterizations of semisimple modules",
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