---
tag: 69HK75YAD1
kind: concept
name: Mori domain
msc: [13F05]
specializes: [0QVD7WBFGN]
sources:
  - url: https://en.wikipedia.org/wiki/Mori_domain
same_as:
  wikidata: Q6912121
  wikipedia: Mori domain
---

## Summary

A Mori domain is an [[integral domain|09S3PFBEPM]] that satisfies the ascending chain condition on divisorial ideals.
Every Noetherian domain and every [[Krull domain|CB9WJCN9BM]] is a Mori domain.
A Mori domain is a Krull domain exactly when it is completely integrally closed.
So Mori domains relate to Krull domains as Noetherian domains relate to Noetherian integrally closed domains.

## Explanations

### L1 human

Divisorial ideals are the ideals that are determined by which principal ideals contain them.
In a Mori domain one cannot build an infinite strictly increasing chain of such ideals.
Principal ideals are divisorial, so in particular there is no infinite strictly increasing chain of principal ideals, and every nonzero nonunit factors into irreducibles.
The ring of all algebraic integers is not a Mori domain: the principal ideals generated by $2^{1/2}, 2^{1/4}, 2^{1/8}, \ldots$ form a strictly increasing chain.