----
> [!definition] Definition. ([[maximal ideal]])
> An [[ideal]] $I \neq \langle 1 \rangle$ of a [[commutative ring|commutative]] [[ring]] $R$ is called a **maximal ideal** if $R / I$ is a [[field]].
>
> The set of maximal ideals of $R$ is sometimes denoted $\text{mSpec }R$.
^definition
> [!equivalence]
> $I$ is a maximal ideal if and only if for all [[ideal|ideals]] $J$ of $R$, $I \subset J \implies (I=J \text{ or } J=R).$
> That is, $I$ is maximal as an ideal iff it is maximal *by inclusion* among proper ideals of $R$.
^equivalence
> [!basicproperties]
> - Evidently, every maximal ideal is a [[prime ideal]] (every [[field]] is an [[integral domain]]);
> - [[prime and maximal ideals align for finite quotients]] (since [[a finite commutative ring is an integral domain iff it is a field]])
> - [[prime iff maximal for nonzero ideals in PID]]
^properties
> [!basicexample]
> - Let $\langle n \rangle$ be an [[ideal]] of $\mathbb{Z}$ (WLOG if we recall that $\mathbb{Z}$ is a [[principal ideal|PID]]). Then in light of property (2) above and the characterization of when $(\mathbb{Z} / n\mathbb{Z})^{\times}$ is a [[field]], we have $\langle n \rangle \text{ prime } \iff \langle n \rangle \text{ maximal} \iff n \text{ prime as an integer}. $
^basic-example
> [!proof]+ Proof of Equivalence.
> [[The correspondence theorem for rings]] identifies $J$ with an [[ideal]] in $R / I$. Using [[division ring iff ideals are {0} and R|commutative ring is a field iff ideals are {0} and R]] we can now say that $I$ is maximal iff $J=I$ or $J=R$.
^proof
----
####
----
#### 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
> ```