Editing Finitely generated module (section)

== Finitely generated modules over a commutative ring ==

For finitely generated modules over a commutative ring ''R'', [[Nakayama's lemma]] is fundamental. Sometimes, the lemma allows one to prove finite dimensional vector spaces phenomena for finitely generated modules. For example, if ''f'' : ''M'' → ''M'' is a [[surjective]] ''R''-endomorphism of a finitely generated module ''M'', then ''f'' is also [[injective function|injective]], and hence is an [[automorphism]] of ''M''.{{sfn|Matsumura|1989|loc=Theorem 2.4}}  This says simply that ''M'' is a [[Hopfian module]]. Similarly, an [[Artinian module]] ''M'' is [[hopfian object|coHopfian]]: any injective endomorphism ''f'' is also a surjective endomorphism.{{sfn|Atiyah|Macdonald|1969|loc=Exercise 6.1}} The [[Forster–Swan theorem]] gives an upper bound for the minimal number of generators of a finitely generated module ''M'' over a commutative Noetherian ring.

Any ''R''-module is an [[inductive limit]] of finitely generated ''R''-submodules. This is useful for weakening an assumption to the finite case (e.g., the [[flat module#Homological algebra|characterization of flatness]] with the [[Tor functor]]).

An example of a link between finite generation and [[integral element]]s can be found in commutative algebras. To say that a commutative algebra ''A'' is a '''finitely generated ring''' over ''R'' means that there exists a set of elements {{nowrap|1=''G'' = {{mset|''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>}}}} of ''A'' such that the smallest subring of ''A'' containing ''G'' and ''R'' is ''A'' itself. Because the ring product may be used to combine elements, more than just ''R''-linear combinations of elements of ''G'' are generated. For example, a [[polynomial ring]] ''R''[''x''] is finitely generated by {{mset|1, ''x''}} as a ring, ''but not as a module''. If ''A'' is a commutative algebra (with unity) over ''R'', then the following two statements are equivalent:{{sfn|Kaplansky|1970|loc=Theorem 17|p=11}}
* ''A'' is a finitely generated ''R'' module.
* ''A'' is both a finitely generated ring over ''R'' and an [[integral element|integral extension]] of ''R''.