# Coherent ring

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{{short description|Algebraic structure}}
In [mathematics](/source/mathematics), a (left) '''coherent ring''' is a [ring](/source/ring_(mathematics)) in which every [finitely generated](/source/Ideal_(ring_theory)) left [ideal](/source/ideal_(ring_theory)) is [finitely presented](/source/finitely_presented_module). 

Many theorems about [finitely generated module](/source/finitely_generated_module)s over [Noetherian ring](/source/Noetherian_ring)s can be extended to finitely presented modules over coherent rings. 

Every left Noetherian ring is left coherent. The ring of [polynomial](/source/polynomial)s 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](/source/if_and_only_if) every [direct product](/source/direct_product) of [flat](/source/flat_module) right [modules](/source/module_(mathematics)) is flat {{harv|Chase|1960}}, {{harv|Anderson|Fuller|1992|p=229}}.  Compare this to: A ring is left Noetherian if and only if every [direct sum](/source/direct_sum_of_modules) of [injective](/source/injective_module) left modules is injective.

==References==
*{{Citation | last1=Anderson | first1=Frank Wylie | last2=Fuller | first2=Kent R | title=Rings and Categories of Modules | publisher=[Springer-Verlag](/source/Springer-Verlag) | location=Berlin, New York | isbn=978-0-387-97845-1 | year=1992}}
*{{Citation | last1=Chase | first1=Stephen U. | title=Direct products of modules | mr=0120260 | year=1960 | journal=[Transactions of the American Mathematical Society](/source/Transactions_of_the_American_Mathematical_Society) | volume=97 | pages=457–473 | doi=10.2307/1993382 | issue=3 | publisher=American Mathematical Society | jstor=1993382| doi-access=free }}
*{{eom|title=Coherent ring|first=V.E.|last= Govorov}}

Category:Ring theory

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