---
tag: 00A2EDFT9Q
kind: definition
name: Principal ideal
msc: [13A15]
defines: 0SVYSP8HCG
uses: [030TZ0DQWT]
sources:
  - url: https://en.wikipedia.org/wiki/Principal_ideal
    locator: Definitions
---

## Definition

Let $R$ be a [[ring|0894J980RK]].
A left ideal $I$ of $R$ is principal if there is an element $a \in R$ such that $I = (a)$, the [[left ideal generated by|030TZ0DQWT]] $\{a\}$.

## Remarks

The left ideal generated by $a$ is $R a = \{r a : r \in R\}$.
If $R$ is commutative, the principal ideal $(a)$ is the set of the multiples of $a$.
Every ideal of $\mathbb{Z}$ is principal: each one is $n\mathbb{Z}$ for some integer $n \ge 0$.
In $\mathbb{Z}[x]$, the ideal $(2, x)$ is not principal.

## Explanations

### L1 human

A principal ideal is the set of all the multiples of one element.
For example, the multiples of $3$ in the integers form the principal ideal $(3)$.