---
tag: 0MYSA6MYNA
kind: definition
name: Ideal
msc: [13A15]
defines: 096Z9MP2ME
uses: [0894J980RK, G5S1M9]
sources:
  - url: https://en.wikipedia.org/wiki/Ideal_(ring_theory)
    locator: Definitions
---

## Definition

Let $R$ be a [[ring|0894J980RK]].
A left ideal of $R$ is a subset $I \subseteq R$ with these properties.

1. $I$ is a subgroup of the additive [[group|G5S1M9]] $(R, +)$.
2. For all $r \in R$ and $a \in I$, $r a \in I$.

If $R$ is a [[commutative ring|0GYGHKX834]], a left ideal is simply called an ideal.

## Remarks

In $\mathbb{Z}$, the set $n\mathbb{Z}$ of the multiples of $n$ is an ideal for every integer $n$.
Every ring $R$ has the ideals $\{0\}$ and $R$.
An ideal contains $1$ if and only if it is all of $R$.
In a noncommutative ring, one also uses right ideals and [[two-sided ideals|0DDA6CY7GJ]].

## Explanations

### L1 human

An ideal is a set that absorbs multiplication.
The even integers are an example: an even number times any integer is even.
Ideals play the role in rings that normal subgroups play in groups.