---
tag: W453BM8T6P
kind: concept
name: Coherent ring
msc: ["13E15", "13E99"]
specializes: [0310REJJY8]
sources:
  - url: https://en.wikipedia.org/wiki/Coherent_ring
same_as:
  wikidata: Q2851410
  wikipedia: Coherent ring
---

## Summary

A coherent ring is a ring in which every finitely generated ideal is [[finitely presented|3EH4C3M5GV]] as a module.
Every [[Noetherian ring|0670T59HGE]] is coherent, but coherence is weaker, and it is kept in many constructions that lose the Noetherian property.
Over a coherent ring, the finitely presented modules form an abelian category, which makes it a good setting for homological algebra without the Noetherian hypothesis.

## Explanations

### L1 human

In a Noetherian ring every ideal has finitely many generators, and the relations among the generators are also finitely generated.
A coherent ring keeps only the second half for finitely generated ideals: once an ideal has finitely many generators, it has finitely many relations among them too.
For example, the polynomial ring $k[x_1, x_2, x_3, \dots]$ in infinitely many variables over a field $k$ is coherent but not Noetherian.