---- > [!definition] Definition. ([[associated graded ring to an ideal]]) > Let $R$ be a ([[commutative ring|commutative]]) [[ring]], $\mathfrak{a} \subset R$ an [[ideal]]. The **associated graded ring** is $G_{\mathfrak{a}}(R):=\bigoplus_{n \geq 0} \frac{\mathfrak{a}^{n}}{\mathfrak{a}^{n+1}} \ \ (\mathfrak{a}^{0}=R)$ > where: > - Addition is defined as usual for a [[direct sum of modules|direct sum of abelian groups]] > - Multiplication is defined on [[graded ring|homogeneous elements]] via $(x + \mathfrak{a}^{n+1}) \cdot (y + \mathfrak{a}^{m+1}):=(xy + \mathfrak{a}^{n+m+1}).$ > > - [ ] properties on page 125 of handwritten notes > [!basicproperties] > ^properties If $R$ is a [[Noetherian ring]], $\mathfrak{a} \subset R$ an [[ideal]], then: 1. $G_{\mathfrak{a}}(R)$ inherits Noetherianity 2. If $M$ is a [[submodule generated by a subset|finitely generated module]] $R$-[[module]] and $(M_{n})$ is a *stable* [[filtration|filtration]] p[d] ---- #### ---- #### 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 > ```