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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0310REJJY8]] and let $I$ be an [[ideal|096Z9MP2ME]] of $R$, with $I^0 = R$.\nThe *associated graded ring* of $R$ with respect to $I$ is the abelian group\n$$\\operatorname{gr}_I(R) = \\bigoplus_{n \\ge 0} I^n / I^{n+1}$$\nwith the following multiplication.\nFor $a \\in I^m$ and $b \\in I^n$ it is\n$$(a + I^{m+1})(b + I^{n+1}) = ab + I^{m+n+1},$$\nand it is extended to all elements by distributivity.\n\n## Remarks\n\nThe product is well defined: if $a' - a \\in I^{m+1}$ and $b' - b \\in I^{n+1}$, then $a'b' - ab = (a' - a) b' + a (b' - b) \\in I^{m+n+1}$.\nWith this product $\\operatorname{gr}_I(R)$ is a commutative [[graded ring|8KB4SQGC0C]] whose degree $0$ part is the [[quotient ring|0076HWHGHQ]] $R/I$.\nExample: for $R = \\Z$ and $I = p\\Z$ with $p$ prime, each $p^n\\Z/p^{n+1}\\Z$ is $\\F_p$ spanned by the class of $p^n$, and $p^m p^n = p^{m+n}$, so $\\operatorname{gr}_I(\\Z) \\cong \\F_p[t]$.\nNon-example: the direct sum $\\bigoplus_{n \\ge 0} I^n$ with the same rule is not $\\operatorname{gr}_I(R)$; it is the [[Rees algebra|W8ZAH8TTHS]] $R[It]$, and $\\operatorname{gr}_I(R)$ is its quotient by $I R[It]$.\nLet $(R, \\mathfrak m)$ be a Noetherian local ring with residue field $k$.\nThen the ring $\\operatorname{gr}_{\\mathfrak m}(R)$ is a polynomial ring over $k$ in $\\dim R$ variables if and only if $R$ is a [[regular local ring|63XXTF6B5M]].\n",
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