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  "body": "\n## Summary\n\nAn invertible ideal of an [[integral domain|0QVD7WBFGN]] $R$ is a fractional ideal $I$ that has an inverse $J$ with $IJ = R$.\nFractional ideals are $R$-submodules of the [[field of fractions|0F2PTGX1A2]] that become ordinary [[ideals|096Z9MP2ME]] after clearing one denominator.\nThe invertible fractional ideals form a group under multiplication, and its quotient by the principal ones is the [[Picard group|DZW58R6T2Q]] of $R$.\nA [[Dedekind domain|7NPY2CN5KB]] is exactly a domain in which every nonzero fractional ideal is invertible.\n\n## Explanations\n\n### L1 human\n\nIdeals can be multiplied, but in general they cannot be divided.\nAn invertible ideal is one that can be divided by: some fractional ideal multiplies it back to the whole ring.\nEvery nonzero principal ideal $(a)$ is invertible, with inverse $(1/a)$.\nIn $\\Z[\\sqrt{-5}]$, the ideal $P = (2, 1 + \\sqrt{-5})$ is not principal, but $P^2 = (2)$, so $\\frac{1}{2} P$ is an inverse of $P$.\n",
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