nLab coherent ring

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Definition

A ring is left coherent if the category of finitely presented left modules is abelian.

For this it is sufficient that finitely generated left ideal of this ring is finitely presented (this is often cited as a definition). Equivalently, a ring is left coherent if and only if it is a left coherent module over itself.

Properties

Every (left) noetherian ring is a coherent ring.

A ring is left coherent iff every finitely presented left module has a resolution by finitely presented projective left modules.

Lombardi & Quitté 2015 argue that coherent rings are more suited to constructive mathematics than noetherian rings. This is probably connected to the importance of coherent sheaves.

References

Last revised on May 21, 2025 at 09:31:04. See the history of this page for a list of all contributions to it.