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  "body": "\n## Definition\n\nLet $R$ be an [[integral domain|0QVD7WBFGN]] that is not a field, and let $K$ be its [[field of fractions|0F2PTGX1A2]].\nThe ring $R$ is a *Dedekind domain* if:\n\n1. $R$ is [[Noetherian|0670T59HGE]];\n2. $R$ is integrally closed: every $x \\in K$ that is a root of a [[monic polynomial|0C3QBCR46M]] with coefficients in $R$ lies in $R$;\n3. every nonzero [[prime ideal|00MHHZ9SNE]] of $R$ is a [[maximal ideal|0PV74TR5FP]].\n\n## Remarks\n\nExample: $\\Z$ is a Dedekind domain.\nIt is a principal ideal domain, hence Noetherian.\nBy the rational root test, a rational root of a monic integer polynomial is an integer.\nEvery nonzero prime ideal $(p)$ is maximal, since $\\Z/(p)$ is a field.\nMore generally, the ring of integers of every [[number field|MWBDFS2KJ5]] is a Dedekind domain, for example $\\Z[\\sqrt{-5}]$, which is not a unique factorisation domain.\nNon-example: $\\Z[\\sqrt{-3}]$ fails clause 2, since $\\omega = (1 + \\sqrt{-3})/2$ lies in its field of fractions, is a root of $x^2 - x + 1$, and is not in $\\Z[\\sqrt{-3}]$.\nAnother non-example: $k[x, y]$ for a field $k$ fails clause 3, since $(x)$ is a nonzero prime ideal contained in the larger ideal $(x, y)$.\nFor a Noetherian integral domain $R$ that is not a field, each of these conditions is equivalent to being Dedekind.\nFirst: every nonzero proper ideal is a product of prime ideals, in a unique way up to order.\nSecond: every nonzero [[fractional ideal|YB4JTB4WJN]] of $R$ is invertible.\nSome authors also count fields as Dedekind domains.\n",
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