Editing Finitely generated module (section)

{{Short description|In algebra, module with a finite generating set}}
In [[mathematics]], a '''finitely generated module''' is a [[module (mathematics)|module]] that has a [[Finite set|finite]] [[Generating set of a module|generating set]]. A finitely generated module over a [[Ring (mathematics)|ring]] ''R'' may also be called a '''finite ''R''-module''', '''finite over ''R''''',<ref>For example, Matsumura uses this terminology.</ref> 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 (mathematics)|field]] is simply a [[Dimension (vector space)|finite-dimensional]] [[vector space]], and a finitely generated module over the [[integer]]s is simply a [[finitely generated abelian group]].