Coherent ring

Summary

A coherent ring is a ring in which every finitely generated ideal is finitely presented as a module. Every Noetherian ring 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.