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  "body": "\n## Definition\n\nLet $(R, \\mathfrak m)$ and $(S, \\mathfrak n)$ be [[local rings|MZ02BKPHJA]].\nA [[ring homomorphism|0VTH8KPCG2]] $f \\colon R \\to S$ is a *local homomorphism* if $f(\\mathfrak m) \\subseteq \\mathfrak n$.\n\n## Remarks\n\nEquivalent forms:\n\n1. $f^{-1}(\\mathfrak n) = \\mathfrak m$;\n2. for every $a \\in R$, if $f(a)$ is a [[unit|0XVEZDE9J8]] of $S$, then $a$ is a unit of $R$.\n\nIndeed $f^{-1}(\\mathfrak n)$ is a proper ideal containing $\\mathfrak m$ when $f(\\mathfrak m) \\subseteq \\mathfrak n$, so it equals $\\mathfrak m$.\nA local homomorphism induces a field homomorphism $R/\\mathfrak m \\to S/\\mathfrak n$ of [[residue fields|NJ3APQ8BHH]].\n\nExample: the quotient map $\\Z_{(p)} \\to \\Z/p\\Z$ is local, since it sends $p\\Z_{(p)}$ to $0$, the maximal ideal of the field $\\Z/p\\Z$.\nFor a prime ideal $\\mathfrak q \\subseteq \\mathfrak p$ of a ring $R$, the natural map $R_{\\mathfrak p} \\to R_{\\mathfrak q}$ is local only when $\\mathfrak q = \\mathfrak p$.\n\nNon-example: the inclusion $\\Z_{(p)} \\to \\Q$ is a ring homomorphism between local rings, but it is not local.\nIt sends $p \\in p\\Z_{(p)}$ to $p$, which is a unit of $\\Q$ and so not in the maximal ideal $(0)$ of $\\Q$.\n",
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