Editing Finitely generated module (section)

== Examples ==
* If a module is generated by one element, it is called a [[cyclic module]].
* Let ''R'' be an [[integral domain]] with ''K'' its [[field of fractions]]. Then every finitely generated ''R''-submodule ''I'' of ''K'' is a [[fractional ideal]]: that is, there is some nonzero ''r'' in ''R'' such that ''rI'' is contained in ''R''. Indeed, one can take ''r'' to be the product of the denominators of the generators of ''I''. If ''R'' is Noetherian, then every fractional ideal arises in this way.
* Finitely generated modules over the ring of [[integer]]s '''Z''' coincide with the [[finitely generated abelian group]]s. These are completely classified by the [[Structure theorem for finitely generated modules over a principal ideal domain|structure theorem]], taking '''Z''' as the principal ideal domain.
* Finitely generated (say left) modules over a [[division ring]] are precisely finite dimensional vector spaces (over the division ring).