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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0310REJJY8]] and let $M$ be an [[$R$-module|0NZ8JA25WK]].\nThe module $M$ is *faithfully flat* if both conditions hold:\n\n1. $M$ is [[flat|73RYYEVDKA]].\n2. For every $R$-module $N$, if $N \\otimes_R M = 0$, then $N = 0$.\n\n## Remarks\n\nThe following are equivalent for an $R$-module $M$:\n\n1. $M$ is faithfully flat.\n2. A sequence $N' \\to N \\to N''$ of $R$-modules is [[exact|JB7ZBFBA9C]] if and only if $N' \\otimes_R M \\to N \\otimes_R M \\to N'' \\otimes_R M$ is exact.\n3. $M$ is flat and $\\mathfrak{m} M \\ne M$ for every [[maximal ideal|0PV74TR5FP]] $\\mathfrak{m}$ of $R$.\n\nExample: a nonzero free module $M = R^{(I)}$ is faithfully flat.\nIt is flat, and $N \\otimes_R R^{(I)} \\cong N^{(I)}$, which is zero only when $N = 0$, since $I$ is not empty.\n\nNon-example: $\\Q$ is a flat $\\Z$-module, so clause 1 holds.\nBut $\\Z/2\\Z \\otimes_\\Z \\Q = 0$ while $\\Z/2\\Z \\ne 0$, since $1 \\otimes q = 2 \\otimes \\frac{q}{2} = 0 \\otimes \\frac{q}{2} = 0$; so clause 2 fails.\n",
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