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  "body": "\n## Summary\n\nA valuation ring is an [[integral domain|0QVD7WBFGN]] $R$ with a simple property, see the [[definition|YVQ9CKGA0T]].\nFor each nonzero $x$ in its [[field of fractions|0F2PTGX1A2]], either $x$ or $1/x$ lies in $R$.\nValuation rings are exactly the rings of elements of nonnegative value for a [[valuation|B85T5VKR3M]].\nThey are [[local rings|MZ02BKPHJA]], and their ideals are totally ordered by inclusion.\n[[Discrete valuation rings|MGM3CNJW9V]] are the Noetherian valuation rings that are not fields.\n\n## Explanations\n\n### L1 human\n\nThink of the fractions whose denominator is not divisible by a prime $p$, such as $3/5$ or $7/2$ for $p = 3$.\nThey form a ring, and any fraction that is not in it, such as $5/3$, has its inverse $3/5$ in it.\nSo the ring takes \"half\" of the fractions in a precise sense: from each pair $x, 1/x$ at least one is in.\nThis half is decided by one rule, the size of the power of $p$, and in general by a valuation.\n",
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