---
tag: 6YCKECNR3T
kind: definition
name: Rees algebra
msc: [13A30]
defines: W8ZAH8TTHS
uses: [0310REJJY8, 096Z9MP2ME, 0SDC934R52, 0GNCRPYXYV, 8KB4SQGC0C]
sources:
  - url: https://en.wikipedia.org/wiki/Rees_algebra
---

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]], let $I$ be an [[ideal|096Z9MP2ME]] of $R$, and let $t$ be a variable, with $I^0 = R$.
The *Rees algebra* of $I$ is the [[subring|0GNCRPYXYV]] of the [[polynomial ring|0SDC934R52]] $R[t]$
$$R[It] = \bigoplus_{n \ge 0} I^n t^n = \set{\textstyle\sum_{n} a_n t^n : a_n \in I^n \text{ for all } n}.$$

## Remarks

It 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 $I^n t^n$.
If $I$ 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.
Example: 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.
Non-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 $I^1 = I$.
The 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.