Finitely generated module

In mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated module over a ring R may also be called a finite R-module, finite over R,^[1] or a module of finite type.

Related concepts include finitely cogenerated modules, finitely presented modules, finitely related modules and coherent modules all of which are defined below. Over a Noetherian ring the concepts of finitely generated, finitely presented and coherent modules coincide.

A finitely generated module over a field is simply a finite-dimensional vector space, and a finitely generated module over the integers is simply a finitely generated abelian group.