---- > [!definition] Definition. ([[graded algebra]]) > A [[graded ring]] $S=\bigoplus_{i}S_{i}$ is a **graded algebra** over a [[graded ring]] $R=\bigoplus_{i}R_{i}$ if it carries an action of $R$ (making it into a 'conventional' $R$-[[algebra]]) and further $R_{i} \cdot S_{j} \subset S_{i+j}$. In particular, if $R=R_{0}$ (so that $R$ is a [[commutative ring]] viewed as a [[graded ring]] by 'concentrating' it in degree $0$) we are just requiring each graded piece of $S$ to be an $R$-[[module]]. ^definition ---- #### Definition from Aluffi. ---- #### 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 > ```