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  "body": "\n## Definition\n\nLet $R$ be a ring and let $M$ be a left [[module|0NZ8JA25WK]] over $R$.\nThe *socle* of $M$, written $\\operatorname{soc}(M)$, is the sum of all [[simple submodules|G00C22AEDH]] of $M$.\nIf $M$ has no simple [[submodule|05WDSN597C]], then $\\operatorname{soc}(M) = 0$.\n\n## Remarks\n\nExample: let $M = \\Z/12\\Z$ as a $\\Z$-module.\nA submodule is simple exactly when it has prime order, so the simple submodules are $6\\Z/12\\Z$ and $4\\Z/12\\Z$, of orders $2$ and $3$.\nTheir sum is $2\\Z/12\\Z$, so $\\operatorname{soc}(\\Z/12\\Z) = 2\\Z/12\\Z$.\n\nNon-example of a nonzero socle: the $\\Z$-module $\\Z$ has socle $0$.\nEach nonzero submodule $n\\Z$ contains the proper nonzero submodule $2n\\Z$, so no submodule of $\\Z$ is simple.\n\nThe socle is a semisimple module, and it contains every semisimple submodule of $M$.\nIt also equals the intersection of all essential submodules of $M$.\nThe socle of the left module ${}_R R$ is a two-sided ideal of $R$.\n",
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