---- > [!definition] Definition. ([[graded ring]]) > A **graded ring** is a (not necessarily commutative) [[ring]] $S$ endowed with a (internal) decomposition $S=\bigoplus_{i \geq 0} S_{i}$ of the [[abelian group]] $(S,+)$ into a [[direct sum of modules|direct sum]] of [[abelian group|abelian groups]] $S_{i}$, for nonnegative integers $i$, such that $S_{i} \cdot S_{j} \subset S_{i+j}.$ Nonzero elements of $S_{i}$ are called **homogeneous of degree $i$**. The condition prescribes that the product of two homogeneous elements $i,j$ is homogeneous, of degree $i+j$. > > Note that $S_0$ is a [[subring]] of $S$, the **grading-zero subring**. [[irrelevant ideal|One has]] $S_{0} \cong \frac{S}{S_{+}}$. ^definition - [ ] justify $S_{0}$ is a [[subring]] (notes, easy) > [!basicexample] > The [[polynomial 4|polynomial ring]] $k[x_{0},\dots,x_{n}]$ admits a grading by degree, by putting $S_{d}:= \{ \text{homogeneous polynomials of total degree }d \}.$ Indeed, every polynomial uniquely decomposes into a sum of [[homogeneous polynomial|homogeneous polynomials]], and the product of a degree-$d$ homogeneous polynomial and a degree-$e$ homogeneous polynomial is a homogeneous polynomial of degree $d+e$. ^basic-example > [!basicproperties] > - $S$ is [[Noetherian ring|Noetherian]] if and only if $S_{0}$ is [[Noetherian ring|Noetherian]] and $S$ is an $S_{0}$-[[algebra]] [[subalgebra generated by a subset|of finite type]]. See [[characterizing Noetherianity in graded rings]]. ^properties ---- #### ---- #### 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 > ```