---
tag: 0RZYDVZ475
kind: definition
name: Principal ideal domain
msc: [13F10]
defines: 0CMD0BCH8T
uses: [09S3PFBEPM, 00A2EDFT9Q]
sources:
  - url: https://en.wikipedia.org/wiki/Principal_ideal_domain
    locator: lead section
---

## Definition

A principal ideal domain is an [[integral domain|09S3PFBEPM]] $R$ in which every ideal is [[principal|00A2EDFT9Q]].
That is, for every ideal $I$ of $R$, there is $a \in R$ with $I = (a)$.

## Remarks

Every [[Euclidean domain is a principal ideal domain|0QKVE1GD9A]]. So $\mathbb{Z}$, $K[x]$ for a field $K$, and $\mathbb{Z}[i]$ are principal ideal domains.
The ring $\mathbb{Z}[x]$ is [[not a principal ideal domain|06E3AG9TM2]].
Every principal ideal domain is a [[unique factorization domain|06YMXJMXYS]].
The ring $\mathbb{Z}[\tfrac{1 + \sqrt{-19}}{2}]$ is a principal ideal domain that is not a Euclidean domain.

## Explanations

### L1 human

In a principal ideal domain, every ideal is the set of the multiples of one element.
For the integers, every ideal is $n\mathbb{Z}$ for some $n$.
This makes divisibility in a principal ideal domain behave much like divisibility of integers.