----- > [!proposition] Proposition. ([[characterizing Noetherianity in graded rings]]) > Let $A=\bigoplus_{n \geq 0}A_{n}$ be a [[graded ring]]. The following are equivalent: >1. $A$ is [[Noetherian ring|Noetherian]]; >2. $A_{0}$ is [[Noetherian ring|Noetherian]] and $A$ is an $A_{0}$-[[algebra]] [[subalgebra generated by a subset|of finite type]]. ^proposition > [!proof]- Proof. ([[characterizing Noetherianity in graded rings]]) > ~ - [ ] todo bring over from notes ----- #### ---- #### 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 > ```