Finitely-generated Module - Finitely Generated Modules Over A Commutative Ring

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 is a surjective R-endomorphism of a finitely generated module M, then f is also injective, and hence is an automorphism of M. This says simply that M is a Hopfian module. Similarly, an Artinian module M is coHopfian: any injective endomorphism f is also a surjective endomorphism.

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 characterization of flatness with the Tor functor.)

An example of a link between finite generation and integral elements 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 G={x₁...x_n} 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 combinations of elements of G are generated. For example, a polynomial ring R is finitely generated by {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:

• A is a finitely generated R module.
• A is both a finitely generated ring over R and an integral extension of R.