----- > [!proposition] Proposition. ([[prime and maximal ideals align for finite quotients]]) > Let $I$ be an [[ideal]] of a [[commutative ring]] $R$. If $R / I$ is *finite*, then $I$ is a [[prime ideal]] if and only if it is a [[maximal ideal]]. > > This is immediate from [[a finite commutative ring is an integral domain iff it is a field]]. ^proposition ----- #### ---- #### 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 > ```