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  "body": "\n## Summary\n\nA Gorenstein ring is a Noetherian ring whose local rings have finite injective dimension as modules over themselves.\nGorenstein rings sit between [[complete intersection rings|EPM2HKGMZS]] and [[Cohen–Macaulay rings|N1QJQEGBFC]]: every complete intersection is Gorenstein and every Gorenstein ring is Cohen–Macaulay.\nThey are exactly the Cohen–Macaulay local rings whose [[canonical module|W5ECXPM6FW]] is the ring itself, which makes duality statements particularly clean.\n\n## Explanations\n\n### L1 human\n\nGorenstein rings have a built-in symmetry.\nFor a finite-dimensional local algebra over a field, the condition says that the \"bottom\" of the ring, the part killed by every non-unit, is one-dimensional.\nThe ring $K[x]/(x^3)$ is Gorenstein: its bottom is spanned by $x^2$.\nThe ring $K[x,y]/(x^2, xy, y^2)$ is not: its bottom is spanned by both $x$ and $y$.\n",
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