---- > [!definition] Definition. ([[finite algebra]]) > An $R$-[[algebra]] $A$ is said to be **finite** if $A$ is [[submodule generated by a subset|finitely generated]] as an $R$-[[module]]. > > We call an inclusion of [[ring|rings]] $A \subset B$ a **finite extension** if $B$ is [[finite algebra|finite]] as an $A$-[[algebra]]. ^definition > [!basicproperties] > - [[transitivity of finiteness and integrality and finite-typedness for algebras]] > - A finite $k$-algebra is both [[Artinian ring|Artinian]] and [[Noetherian ring|Noetherian]] as a ring: since it is a finite-dimensional [[vector space]], any chain of subspaces has finite length. As corollaries, we get things like: every [[prime ideal]] of a finite $k$-algebra is [[maximal ideal|maximal]], and $\text{Spec }A$ is finite (in fact, bounded in terms of $\text{dim}_{k}A$) as... ^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 > ```