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  "body": "\n## Summary\n\nA Dedekind domain is an integral domain in which every nonzero proper ideal is, in a unique way, a product of prime ideals.\nRings of integers of [[number fields|MWBDFS2KJ5]] are Dedekind domains, and so are all [[principal ideal domains|0CMD0BCH8T]].\nIn such rings unique factorisation of elements may fail, but unique factorisation of ideals always holds.\nThe [[ideal class group|WQ46BD3ARY]] measures the gap between the two.\n\n## Explanations\n\n### L1 human\n\nIn the integers every number splits into primes in only one way.\nIn larger number rings this can fail: in $\\Z[\\sqrt{-5}]$, $6 = 2 \\cdot 3 = (1 + \\sqrt{-5})(1 - \\sqrt{-5})$ are two different factorisations into irreducible elements.\nDedekind's repair was to factor ideals instead of numbers.\nIn a Dedekind domain every nonzero proper ideal splits into prime ideals in exactly one way, and the two factorisations of $6$ come from regrouping the same four prime ideals.\n",
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