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  "body": "\n## Summary\n\nA local ring is a [[commutative ring|0310REJJY8]] with exactly one [[maximal ideal|0PV74TR5FP]].\nEquivalently, its non-[[units|0XVEZDE9J8]] form an ideal.\nLocal rings arise as [[localizations at prime ideals|X6JD4AG973]] and describe the behaviour of a ring, or of a geometric space, near one point.\nMany theorems of commutative algebra are first proved for local rings, where [[Nakayama's lemma|15JHBYR6KH]] is a strong tool.\n\n## Explanations\n\n### L1 human\n\nA local ring is a ring that \"sees only one point\".\nA typical example is the ring of rational numbers whose denominators are odd: an element is invertible exactly when its numerator is odd too.\nThe elements that are not invertible, those with even numerator, form one ideal, the maximal ideal.\nAnother example is the ring of formal power series $k[\\![x]\\!]$, where a series is invertible exactly when its constant term is nonzero.\nThe [[definition|WJS9TPSGVM]] gives the condition and its equivalent forms.\n",
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