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  "body": "\n## Summary\n\nA Krull domain is an [[integral domain|09S3PFBEPM]] that is the intersection of a family of discrete valuation rings of its fraction field.\nThe family must satisfy a finiteness condition: each nonzero element has nonzero valuation for only finitely many of them.\n[[Unique factorisation domains|06YMXJMXYS]], Dedekind domains and Noetherian [[integrally closed domains|H20P600HQS]] are Krull domains.\nKrull domains are the natural setting for the [[divisor class group|WN8PK6RQZR]], which measures their failure to be factorial.\n\n## Explanations\n\n### L1 human\n\nIn $\\Z$, a rational number is an integer exactly when no prime appears in its denominator.\nFor each prime $p$, the exponent of $p$ in a fraction is a \"valuation\", and $\\Z$ is the set of fractions where all these exponents are $\\ge 0$.\nA Krull domain is a ring that can be described in the same way by a family of such exponent functions.\nEach nonzero element must have nonzero exponent for only finitely many of them.\nThis gives a theory of divisibility by \"prime divisors\" even when elements do not factor uniquely.\n",
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