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  "body": "\n## Definition\n\nLet $R$ be a commutative ring and $M$ an $R$-module.\nThe *injective dimension* $\\operatorname{id}_R M$ is the least $n \\ge 0$ such that there is an exact sequence $0 \\to M \\to I^0 \\to \\cdots \\to I^n \\to 0$ of $R$-modules.\nHere every $I^j$ must be an [[injective module|85H7PRHDPZ]], and $\\operatorname{id}_R M = \\infty$ if there is no such sequence.\nA commutative [[Noetherian|0670T59HGE]] [[local ring|MZ02BKPHJA]] $R$ is a *Gorenstein local ring* if $\\operatorname{id}_R R \u003c \\infty$.\nA commutative Noetherian ring $R$ is a *Gorenstein ring* if the [[localization|X6JD4AG973]] $R_{\\mathfrak{p}}$ is a Gorenstein local ring for every [[prime ideal|00MHHZ9SNE]] $\\mathfrak{p}$.\n\n## Remarks\n\nA Gorenstein local ring is [[Cohen–Macaulay|N1QJQEGBFC]], and then $\\operatorname{id}_R R = \\dim R$.\nEquivalently, a Noetherian local ring $(R, \\mathfrak{m}, k)$ of dimension $d$ is Gorenstein if and only if $\\operatorname{Ext}^i_R(k, R) = 0$ for $i \\ne d$ and $\\operatorname{Ext}^d_R(k, R) \\cong k$.\nFor an Artinian local ring this says that the socle $(0 :_R \\mathfrak{m})$ is a $k$-vector space of dimension $1$, or that $R$ is the [[injective hull|QCJN05GN64]] of $k$ as an $R$-module.\nExample: $R = K[x]/(x^n)$ with $n \\ge 1$ and $K$ a field is Gorenstein, since its socle is $K x^{n-1}$.\nNon-example: $R = K[x,y]/(x^2, xy, y^2)$ is Artinian local with socle $Kx \\oplus Ky$ of dimension $2$, so it is not Gorenstein, although it is Cohen–Macaulay.\n",
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