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  "body": "\n## Definition\n\nLet $R$ be an [[integral domain|0QVD7WBFGN]] and $K$ its [[field of fractions|0F2PTGX1A2]].\nA *fractional ideal* of $R$ is a subset $I \\subseteq K$ such that:\n\n1. $I$ is an $R$-[[submodule|05WDSN597C]] of $K$;\n2. $I \\ne \\set{0}$;\n3. there is a nonzero $r \\in R$ with $rI \\subseteq R$.\n\nThe product of fractional ideals $I$ and $J$ is the set $IJ$ of finite sums $\\sum x_k y_k$ with $x_k \\in I$ and $y_k \\in J$.\nA fractional ideal $I$ is *invertible* if $IJ = R$ for some fractional ideal $J$.\n\n## Remarks\n\nExample: $I = \\tfrac12 \\Z \\subseteq \\Q$ for $R = \\Z$; it is a nonzero $\\Z$-submodule, and $2I = \\Z \\subseteq \\Z$, so it is a fractional ideal, with inverse $2\\Z$.\nNon-example: $\\Z[\\tfrac12] = \\set{a/2^k : a \\in \\Z, k \\ge 0}$ is a nonzero $\\Z$-submodule of $\\Q$, but it fails clause 3: for any nonzero integer $r$, choose $k$ with $2^k \u003e \\abs{r}$; then $r/2^k \\in r\\Z[\\tfrac12]$ is not an integer.\nEvery nonzero [[ideal|096Z9MP2ME]] of $R$ is a fractional ideal, with $r = 1$; these are the integral ideals.\nFor $x \\in K^{\\times}$, the set $xR$ is a principal fractional ideal, with inverse $x^{-1}R$.\nWhen $R$ is Noetherian, clause 3 is equivalent to $I$ being a finitely generated $R$-module.\nSome authors, and Mathlib, also count $\\set{0}$ as a fractional ideal.\n",
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