---
tag: 030TZ0DQWT
kind: definition
name: Ideal generated by a subset
msc: [13A15]
defines: 0GSW76Q14N
uses: [0MYSA6MYNA, 0GYGHKX834]
sources:
  - url: https://en.wikipedia.org/wiki/Ideal_(ring_theory)
    locator: Definitions
---

## Definition

Let $R$ be a [[ring|0894J980RK]] and let $X \subseteq R$ be a subset.
The left ideal generated by $X$ is the intersection of all the left [[ideals|0MYSA6MYNA]] of $R$ that contain $X$.
The notation for it is $(X)$.
It is the smallest left ideal of $R$ that contains $X$.

## Remarks

The left ideal $(X)$ is the set of the finite sums $r_1 x_1 + \dots + r_n x_n$ with $n \ge 0$, $r_i \in R$ and $x_i \in X$.
For a finite set, one writes $(x_1, \dots, x_n)$ for $(\{x_1, \dots, x_n\})$.
In $\mathbb{Z}$, the ideal $(4, 6)$ is $2\mathbb{Z}$.
An ideal generated by one element is a [[principal ideal|00A2EDFT9Q]].

## Explanations

### L1 human

Start with some elements of the ring.
The ideal that they generate contains all the sums of their multiples, and nothing else.
In the integers, the multiples of $4$ and $6$ give all the even numbers, because $2 = 2 \cdot 4 - 1 \cdot 6$.