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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0310REJJY8]], let $I$ be an [[ideal|096Z9MP2ME]] of $R$, and let $t$ be a variable, with $I^0 = R$.\nThe *Rees algebra* of $I$ is the [[subring|0GNCRPYXYV]] of the [[polynomial ring|0SDC934R52]] $R[t]$\n$$R[It] = \\bigoplus_{n \\ge 0} I^n t^n = \\set{\\textstyle\\sum_{n} a_n t^n : a_n \\in I^n \\text{ for all } n}.$$\n\n## Remarks\n\nIt is a subring because $1 \\in I^0$ and $I^m I^n \\subseteq I^{m+n}$, and it is a [[graded ring|8KB4SQGC0C]] with degree $n$ part $I^n t^n$.\nIf $I$ is generated by $a_1, \\dots, a_s$, then $R[It]$ is generated as an $R$-algebra by $a_1 t, \\dots, a_s t$; so if $R$ is Noetherian, then $R[It]$ is Noetherian.\nExample: for $R = k[x]$, with $k$ a field, and $I = (x)$, one gets $R[It] = k[x, xt]$, and $x$ and $xt$ are algebraically independent, so this is a polynomial ring in two variables.\nNon-example: for a proper ideal $I$, the whole ring $R[t]$ is not $R[It]$, because $t$ lies in $R[t]$ but its coefficient $1$ does not lie in $I^1 = I$.\nThe quotient $R[It]/I R[It]$ is the [[associated graded ring|8HFAEGFQT1]] $\\bigoplus_{n} I^n/I^{n+1}$, and the extended Rees algebra $R[It, t^{-1}]$ is used as well.\n",
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