# Maximal ideal

A (two-sided) proper [[ideal]] $I \triangleleft R$ is called a **maximal ideal** iff whenever $J \trianglelefteq R$ is an [[ideal]] such that $I \trianglelefteq J \trianglelefteq R$,
then either $I = J$ or $J = R$. #m/def/ring 
$$
\begin{align*}
I \trianglelefteq J \trianglelefteq R \iff [I = J] \lor [J = R]
\end{align*}
$$
Maximal ideal generalizes [[Irreducible element]] in the same way that [[Prime ideal]] generalized [[Prime element]].


## Properties

- [[A maximal ideal in a commutative ring is prime]] 
- [[Condition for a quotient commutative ring to be a field]]
- [[Every commutative ring has a maximal ideal]]

#
---
#state/tidy | #lang/en | #SemBr