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  "body": "\n## Definition\n\nLet $R$ be a ring with identity element and let $M$ be a left [[module|0NZ8JA25WK]] over $R$.\nA *submodule* of $M$ is a subset $N \\subseteq M$ such that:\n\n1. $N$ is a subgroup of the abelian group $(M, +)$.\n2. $rx \\in N$ for all $r \\in R$ and $x \\in N$.\n\n## Remarks\n\nA submodule is a module over $R$ with the restricted operations.\nThe submodules of $R$, as a left module over itself, are the left ideals of $R$.\nThe $\\Z$-submodules of an abelian group are its subgroups, and over a field the submodules are the [[subspaces|00YC1D652K]].\nIntersections and sums of submodules are submodules.\n\n## Explanations\n\n### L1 human\n\nA submodule is a module inside a bigger module.\nIt contains the sums and the scalar multiples of its elements.\nFor example, the even integers are a submodule of $\\Z$ over $\\Z$.\n",
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