Coherent ring

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title: "Coherent ring" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["ring-theory"] description: "Algebraic structure" topic_path: "general/ring-theory" source: "https://en.wikipedia.org/wiki/Coherent_ring" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Algebraic structure ::

In mathematics, a (left) coherent ring is a ring in which every finitely generated left ideal is finitely presented.

Many theorems about finitely generated modules over Noetherian rings can be extended to finitely presented modules over coherent rings.

Every left Noetherian ring is left coherent. The ring of polynomials in an infinite number of variables over a left Noetherian ring is an example of a left coherent ring that is not left Noetherian.

A ring is left coherent if and only if every direct product of flat right modules is flat , . Compare this to: A ring is left Noetherian if and only if every direct sum of injective left modules is injective.

References

::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::