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 finitely generated -submodule is finitely presented (equivalently: such that the kernel of any (not neccessarily surjective) linear map is finitely generated).
Coherent modules behave well over noetherian rings and to some extent generally over coherent rings.
A geometric globalization of a notion of coherent module is a notion of a coherent sheaf of -modules for a ringed space .
B. Kaup, Coherent D-modules, pp. 109–270 in “Algebraic D-modules”, A. Borel ed., Academic Press.