---
tag: 0W4X9G6263
kind: concept
name: Unique factorisation domain
aliases: [unique-factorization-domain, ufd, factorial-ring]
msc: [13F15]
sources:
  - url: https://en.wikipedia.org/wiki/Unique_factorization_domain
    locator: lead section
same_as:
  wikidata: Q1052579
  wikipedia: Unique factorization domain
---

## Summary

A unique factorisation domain is an integral domain in which every nonzero nonunit is a product of irreducible elements, in an essentially unique way.
The fundamental theorem of arithmetic says that $\mathbb{Z}$ is one.
The [[definition of a unique factorisation domain|06YMXJMXYS]] gives the conditions.