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  "body": "\n## Definition\n\nA *local ring* is a [[commutative ring|0310REJJY8]] $R$ that has exactly one [[maximal ideal|0PV74TR5FP]] $\\mathfrak m$.\nOne writes $(R, \\mathfrak m)$, or $(R, \\mathfrak m, k)$ with the [[residue field|NJ3APQ8BHH]] $k = R/\\mathfrak m$.\n\n## Remarks\n\nFor a commutative ring $R$, the following are equivalent:\n\n1. $R$ is local;\n2. $R \\ne 0$ and the set of non-[[units|0XVEZDE9J8]] of $R$ is an ideal;\n3. $R \\ne 0$ and, for every $a \\in R$, $a$ or $1 - a$ is a unit.\n\nIn a local ring, the maximal ideal is exactly the set of non-units, because every non-unit lies in some maximal ideal.\n\nExample: let $p$ be a prime and $\\Z_{(p)}$ the ring of rationals $a/b$ with $p \\nmid b$.\nThe element $a/b$ in lowest terms is a unit exactly when $p \\nmid a$.\nSo the non-units are the elements of $p\\Z_{(p)}$, which is an ideal, and $\\Z_{(p)}$ is local.\nOther examples are every field, the ring $\\Z/p^n\\Z$, the formal power series ring $k[\\![x]\\!]$ over a field $k$, and every [[localization at a prime ideal|X6JD4AG973]].\n\nNon-example: $\\Z$ is not local, because $2\\Z$ and $3\\Z$ are two different maximal ideals.\nEquivalently, $3$ and $1 - 3 = -2$ are both non-units of $\\Z$.\nThe zero ring is not local either, since it has no maximal ideal.\n\nSome authors call a Noetherian local ring a local ring, and call the general case a quasi-local ring.\n",
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