---
tag: 09J6KGT65G
kind: definition
name: Maximal ideal
msc: [13A15]
defines: 0PV74TR5FP
uses: [0MYSA6MYNA]
sources:
  - url: https://en.wikipedia.org/wiki/Maximal_ideal
    locator: Definition
---

## Definition

Let $R$ be a [[ring|0894J980RK]].
An [[ideal|0MYSA6MYNA]] $M$ of $R$ is maximal if it has these two properties.

1. $M \neq R$.
2. For every ideal $I$ of $R$ with $M \subseteq I$, either $I = M$ or $I = R$.

## Remarks

The definition is the same for left, right and two-sided ideals, with ideals of the same type in condition 2.
In $\mathbb{Z}$, the ideal $n\mathbb{Z}$ with $n \ge 0$ is maximal if and only if $n$ is a prime number.
In a [[commutative ring|0GYGHKX834]] $R$, an ideal $M$ is maximal if and only if [[$R/M$ is a field|0WD3SE7S1X]].
The zero ideal of a [[field|0QA8CHH8MG]] is its only maximal ideal.

## Explanations

### L1 human

A maximal ideal is as large as possible without being the whole ring.
In the integers, $(6)$ is not maximal, because $(6) \subseteq (2)$.
The ideal $(2)$ is maximal, because no ideal lies strictly between $(2)$ and $\mathbb{Z}$.