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  "body": "\n## Definition\n\nLet $(R, \\mathfrak{m})$ be a [[local ring|MZ02BKPHJA]].\nA *Henselization* of $R$ is a local ring $R^h$ together with a [[local homomorphism|P8YTMGEN9R]] $i \\colon R \\to R^h$ such that:\n\n1. $R^h$ is a [[Henselian local ring|RNXPV3KC5J]].\n2. For every Henselian local ring $S$ and every local homomorphism $f \\colon R \\to S$, there is exactly one local homomorphism $g \\colon R^h \\to S$ with $g \\circ i = f$.\n\n## Remarks\n\nA Henselization exists for every local ring, and the universal property makes it unique up to a unique isomorphism.\nOne construction takes the direct limit of the localizations of [[étale|FTRW0679E2]] $R$-algebras at primes over $\\mathfrak{m}$ with trivial residue field extension.\nThe map $i$ is [[faithfully flat|C31PYYPRGC]], the ideal $\\mathfrak{m} R^h$ is the maximal ideal of $R^h$, and $i$ induces an isomorphism of residue fields.\nIf $R$ is Noetherian, then $R^h$ is Noetherian and has the same [[completion|KBN13NPYJQ]] as $R$.\n\nExample: if $R$ is already Henselian, then $R$ with the identity map is a Henselization, because $g = f$ is the only choice in clause 2.\n\nNon-example: $\\Z_{(5)}$ with the identity map is not its own Henselization, because it fails clause 1.\nThe polynomial $x^2 - 6$ has the simple root $1$ modulo $5$, but $6$ is not a square in $\\Q$, so the root does not lift to $\\Z_{(5)}$.\n",
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