Suppose we are given a (not neccesarily commutative) unital ring . A left -module is finitely generated if there is an exact sequence of left -modules where is a natural number. is a noetherian -module if each -submodule is finitely generated. A ring is noetherian if it is noetherian as a left -module.
A left -module is finitely presented (or of finite presentation) if there exists an exact sequence where are natural numbers. A left coherent module is a left -module which is finitely generated and such that every -submodule is finitely presented.
Coherent modules behave well over noetherian rings.
A geometric globalization of a notion of coherent module is a notion of a coherent sheaf of -modules for a ringed space .