---
tag: 055HT85T3S
kind: statement
role: theorem
name: PID implies UFD
msc: [13F15]
uses: [0RZYDVZ475, 06YMXJMXYS]
sources:
  - url: https://en.wikipedia.org/wiki/Unique_factorization_domain
    locator: Examples
---

## Statement

Let $R$ be a [[principal ideal domain|0RZYDVZ475]].
Then $R$ is a [[unique factorisation domain|06YMXJMXYS]].

## Remarks

Existence of factorisations: a principal ideal domain is Noetherian, so a process of splitting an element into factors must stop.
Uniqueness: in a principal ideal domain, an irreducible element generates a maximal ideal, so it is prime.
Consequences: $\mathbb{Z}$, $\mathbb{Z}[i]$ and $K[x]$ for a field $K$ have unique factorisation.
The converse is false: $\mathbb{Z}[x]$ is a unique factorisation domain but [[not a principal ideal domain|06E3AG9TM2]].

## Explanations

### L1 human

In a principal ideal domain, every irreducible element behaves like a prime number.
That is the key to unique factorisation.
So every Euclidean domain, such as the integers, has unique factorisation.