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.
A finite generating set need not be a basis, since it need not be linearly independent over R. What is true is: M is finitely generated if and only if there is a surjective R-linear map: for some n (M is a quotient of a free module of finite rank).
However, it may occur that S does not contain any finite generating set of minimal cardinality.
In the case where the module M is a vector space over a field R, and the generating set is linearly independent, n is well-defined and is referred to as the dimension of M (well-defined means that any linearly independent generating set has n elements: this is the dimension theorem for vector spaces).
Any module is the union of the directed set of its finitely generated submodules.
A module M is finitely generated if and only if any increasing chain Mi of submodules with union M stabilizes: i.e., there is some i such that Mi = M. This fact with Zorn's lemma implies that every nonzero finitely generated module admits maximal submodules.
Consider the submodule K consisting of all those polynomials with zero constant term.
In general, a module is said to be Noetherian if every submodule is finitely generated.
Let B be a ring and A its subring such that B is a faithfully flat right A-module.
Then a left A-module F is finitely generated (resp.
[2] The Forster–Swan theorem gives an upper bound for the minimal number of generators of a finitely generated module M a commutative Noetherian ring.
For finitely generated modules over a commutative ring R, Nakayama's lemma is fundamental.
[4] 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, ..., xn} of A such that the smallest subring of A containing G and R is A itself.
If A is a commutative algebra (with unity) over R, then the following two statements are equivalent:[5] Let M be a finitely generated module over an integral domain A with the field of fractions K. Then the dimension
This number is the same as the number of maximal A-linearly independent vectors in M or equivalently the rank of a maximal free submodule of M (cf.
When A is Noetherian, by generic freeness, there is an element f (depending on M) such that
Then the rank of this free module is the generic rank of M. Now suppose the integral domain A is generated as algebra over a field k by finitely many homogeneous elements of degrees
is the generic rank of M.[6] A finitely generated module over a principal ideal domain is torsion-free if and only if it is free.
But it can also be shown directly as follows: let M be a torsion-free finitely generated module over a PID A and F a maximal free submodule.
The following conditions are equivalent to M being finitely generated (f.g.): From these conditions it is easy to see that being finitely generated is a property preserved by Morita equivalence.
The conditions are also convenient to define a dual notion of a finitely cogenerated module M. The following conditions are equivalent to a module being finitely cogenerated (f.cog.
This is easily seen by applying the characterization using the finitely generated essential socle.
For example, an infinite direct product of nonzero rings is a finitely generated (cyclic!)
module over itself, however it clearly contains an infinite direct sum of nonzero submodules.
Finitely generated modules do not necessarily have finite co-uniform dimension either: any ring R with unity such that R/J(R) is not a semisimple ring is a counterexample.
Some crossover occurs for projective or flat modules.
It is true also that the following conditions are equivalent for a ring R: Although coherence seems like a more cumbersome condition than finitely generated or finitely presented, it is nicer than them since the category of coherent modules is an abelian category, while, in general, neither finitely generated nor finitely presented modules form an abelian category.