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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 *flat* if the following holds for every injective $R$-[[linear map|0NRYHJPQG2]] $f \\colon N' \\to N$ of $R$-modules.\nThe map\n$$f \\otimes \\id_M \\colon N' \\otimes_R M \\to N \\otimes_R M$$\nis injective, where $\\otimes_R$ is the [[tensor product|F0HQXQ0HCD]].\n\n## Remarks\n\nSince $- \\otimes_R M$ is always right exact, $M$ is flat exactly when $- \\otimes_R M$ takes [[exact sequences|JB7ZBFBA9C]] to exact sequences.\nIt is enough to check the condition for the inclusions $I \\to R$ of finitely generated [[ideals|096Z9MP2ME]]: $M$ is flat if and only if $I \\otimes_R M \\to M$ is injective for each such $I$.\nOver a [[principal ideal domain|0CMD0BCH8T]], a module is flat exactly when it is torsion-free.\nProjective modules are flat, and a finitely presented flat module is projective.\n\nExample: $\\Q$ is a flat $\\Z$-module.\nFor any $\\Z$-module $N$, the module $N \\otimes_\\Z \\Q$ is the localization $S^{-1}N$ with $S = \\Z \\setminus \\set{0}$, and localization preserves injective maps.\n\nNon-example: $\\Z/2\\Z$ is not a flat $\\Z$-module.\nThe map $f \\colon \\Z \\to \\Z$, $x \\mapsto 2x$, is injective, but $f \\otimes \\id$ is multiplication by $2$ on $\\Z \\otimes_\\Z \\Z/2\\Z \\cong \\Z/2\\Z$, which is the zero map on a nonzero module.\n",
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