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  "body": "\n## Definition\n\nLet $P$ be a property of modules over commutative rings.\nThe property $P$ is *local* if, for every [[commutative ring|0310REJJY8]] $R$ and every $R$-[[module|0NZ8JA25WK]] $M$, the following are equivalent:\n\n1. $M$ has $P$ as an $R$-module;\n2. for every [[prime ideal|00MHHZ9SNE]] $\\mathfrak p$ of $R$, the [[localized module|68E0F8N94E]] $M_{\\mathfrak p}$ has $P$ as an $R_{\\mathfrak p}$-module.\n\nA property of commutative rings is *local* in the same way, with $R$ in place of $M$ and the [[localizations|X6JD4AG973]] $R_{\\mathfrak p}$ in place of $M_{\\mathfrak p}$.\nA property of module homomorphisms $\\varphi \\colon M \\to N$ is local in the same way, with the localized maps $\\varphi_{\\mathfrak p} \\colon M_{\\mathfrak p} \\to N_{\\mathfrak p}$.\n\n## Remarks\n\nExample: the property \"$M = 0$\" is local.\nIf $M \\ne 0$, pick $m \\ne 0$ in $M$; its [[annihilator|HB4XDKJTW8]] is a proper ideal, so it lies in a [[maximal ideal|0PV74TR5FP]] $\\mathfrak m$.\nThen $m/1 \\ne 0$ in $M_{\\mathfrak m}$, since $um \\ne 0$ for all $u \\notin \\mathfrak m$.\nThe converse direction is clear.\nIt follows that injectivity and surjectivity of a module map, exactness of a sequence, and [[flatness|73RYYEVDKA]] are local.\n\nIn each case it is enough to check the maximal ideals instead of all prime ideals.\n\nNon-example: being an [[integral domain|0QVD7WBFGN]] is not a local property of rings.\nFor a field $k$, the ring $R = k \\times k$ has the two prime ideals $k \\times 0$ and $0 \\times k$, and both localizations are isomorphic to $k$, a domain.\nBut $(1, 0) \\cdot (0, 1) = 0$ in $R$, so $R$ is not a domain.\nBeing [[Noetherian|0670T59HGE]] is also not local; see [[localization of a Noetherian ring is Noetherian|C618HAK3X6]].\n",
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