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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 subset $E \\subseteq M$ is a *basis* of $M$ when:\n\n1. Each $x \\in M$ is a finite sum $r_1 e_1 + \\cdots + r_k e_k$ with $r_i \\in R$ and $e_i \\in E$.\n2. If $e_1, \\dots, e_k \\in E$ are distinct and $r_1 e_1 + \\cdots + r_k e_k = 0$, then $r_1 = \\cdots = r_k = 0$.\n\nThe module $M$ is *free* when it has a basis.\n\n## Remarks\n\n$R^n$ is free, with the standard basis; each vector space is free over its field.\nFor $n \\ge 2$, the $\\Z$-module $\\Z / n\\Z$ is not free, because $n x = 0$ for all its elements.\nOver a nonzero commutative ring, any two bases of a free module have the same cardinality, called the *rank*.\nA submodule of a free module need not be free, but over a principal ideal domain it is free.\n\n## Explanations\n\n### L1 human\n\nA free module is a module that behaves like $R^n$: it has a basis.\nEvery vector space has a basis, but many modules do not.\nFor example, $\\Z / 2\\Z$ has no basis over $\\Z$, because $2x = 0$ for every element $x$.\n",
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