---- $R$ is a ([[commutative ring|commutative]]) [[ring]]. > [!definition] Definition. ([[Rees ring]]) > Let $\mathfrak{a}$ be an [[ideal]] of $R$. Let $M$ be an $R$-[[module]], and $(M_{n})_{n \geq 0}$ a [[filtration|stable]] $\mathfrak{a}$-[[filtration|filtration]] of $M$. We define a [[graded ring]] $R^{*}$, called the **Rees ring**, which is like the [[associated graded ring to an ideal|associated graded ring]] to $\mathfrak{a}$ but we don't do any [[quotient ring|dividing]]. In particular: $R^{*}:= \bigoplus_{n \geq 0} \mathfrak{a}^{n}, \text{ recalling }\mathfrak{a}^{0}=R$ We define also a [[graded module|graded]] $R^{*}$-[[module]] $M^{*}$ as[^1] $M^{*}:= \bigoplus_{n \geq 0}M_{n}.$ Note that $R^{*}$ is [[Noetherian ring|Noetherian]] if $R$ is: if $\mathfrak{a}=\langle x_{1},\dots,x_{r} \rangle$ is [[submodule generated by a subset|finitely generated]], then $R^{*}$ is [[subalgebra generated by a subset|generated]] as an $R=\mathfrak{a}^{0}$-[[algebra]] by $\{ x_{1},\dots,x_{r} \}$, hence is [[Noetherian ring|Noetherian]] by [[Hilbert's basis theorem]]. ^definition ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```