Finitely generated module

Definition

The left R-module M is finitely generated if there exist a₁, a₂, ..., a_n in M such that for all x in M, there exist r₁, r₂, ..., r_n in R with x = r₁a₁ + r₂a₂ + ... + r_na_n.

The set {a₁, a₂, ..., a_n} is referred to as a generating set for M in this case. The finite generators need not be a basis, since they 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.)

If a set S generates a module that is finitely generated, then the finite generators of the module can be taken from S at the expense of possibly increasing the number of the generators (since only finitely many elements in S are needed to express the finite generators).

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 M_i of submodules with union M stabilizes: i.e., there is some i such that M_i = M. If any increasing chain of submodules stabilizes (i.e., any submodule is finitely generated), then the module M is called a Noetherian module.

Examples

Some facts

Every homomorphic image of a finitely generated module is finitely generated. In general, submodules of finitely generated modules need not be finitely generated. As an example, consider the ring R = Z[X₁, X₂, ...] of all polynomials in countably many variables. R itself is a finitely generated R-module (with {1} as generating set). Consider the submodule K consisting of all those polynomials with zero constant term. Since every polynomial contains only finitely many terms whose coefficients are non-zero, the R-module K is not finitely generated.

In general, a module is said to be Noetherian if every submodule is finitely generated. A finitely generated module over a Noetherian ring is a Noetherian module (and indeed this property characterizes Noetherian rings): A module over a Noetherian ring is finitely generated if and only if it is a Noetherian module. This resembles, but is not exactly Hilbert's basis theorem, which states that the polynomial ring R[X] over a Noetherian ring R is Noetherian. Both facts imply that a finitely generated commutative algebra over a Noetherian ring is again a Noetherian ring.

More generally, an algebra (e.g., ring) that is a finitely generated module is a finitely generated algebra. Conversely, if a finitely generated algebra is integral (over the coefficient ring), then it is finitely generated module. (See integral element for more.)

Let 0 → M′ → M → M′′ → 0 be an exact sequence of modules. Then M is finitely generated if M′, M′′ are finitely generated. There are some partial converses to this. If M is finitely generated and M'' is finitely presented (which is stronger than finitely generated; see below), then M′ is finitely generated. Also, M is Noetherian (resp. Artinian) if and only if M′, M′′ are Noetherian (resp. Artinian).

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. finitely presented) if and only if the B-module B ⊗_A F is finitely generated (resp. finitely presented).

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, 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-linear combinations of elements of G are generated. For example, a polynomial ring R[x] 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:

Generic rank

Let M be a finitely generated module over an integral domain A with the field of fractions K. Then the dimension dim K ⁡ ( M ⊗ A K ) is called the generic rank of M over A. 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. rank of an abelian group.) Since ( M / F ) ( 0 ) = M ( 0 ) / F ( 0 ) = 0 , M / F is a torsion module. When A is Noetherian, by generic freeness, there is an element f (depending on M) such that M [ f − 1 ] is a free A [ f − 1 ] -module. 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 d i . Suppose M is graded as well and let P M ( t ) = ∑ dim k ⁡ ( M n ) t n be the Poincaré series of M. By the Hilbert–Serre theorem, there is a polynomial F such that P M ( t ) = F ( t ) ∏ ( 1 − t d i ) − 1 . Then F ( 1 ) is the generic rank of M.

A finitely generated module over a principal ideal domain is torsion-free if and only if it is free. This is a consequence of the structure theorem for finitely generated modules over a principal ideal domain, the basic form of which says a finitely generated module over a PID is a direct sum of a torsion module and a free module. 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. Let f be in A such that f M ⊂ F . Then f M is free since it is a submodule of a free module and A is a PID. But now f : M → f M is an isomorphism since M is torsion-free.

By the same argument as above, a finitely generated module over a Dedekind domain A (or more generally a semi-hereditary ring) is torsion-free if and only if it is projective; consequently, a finitely generated module over A is a direct sum of a torsion module and a projective module. A finitely generated projective module over a Noetherian integral domain has constant rank and so the generic rank of a finitely generated module over A is the rank of its projective part.

Equivalent definitions and finitely cogenerated modules

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.):

Both f.g. modules and f.cog. modules have interesting relationships to Noetherian and Artinian modules, and the Jacobson radical J(M) and socle soc(M) of a module. The following facts illustrate the duality between the two conditions. For a module M:

Finitely cogenerated modules must have finite uniform dimension. This is easily seen by applying the characterization using the finitely generated essential socle. Somewhat asymmetrically, finitely generated modules do not necessarily have finite uniform dimension. 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.

Finitely presented, finitely related, and coherent modules

Another formulation is this: a finitely generated module M is one for which there is an epimorphism

Suppose now there is an epimorphism,

for a module M and free module F.

Over any ring R, coherent modules are finitely presented, and finitely presented modules are both finitely generated and finitely related. For a Noetherian ring R, finitely generated, finitely presented, and coherent are equivalent conditions on a module.

Some crossover occurs for projective or flat modules. A finitely generated projective module is finitely presented, and a finitely related flat module is projective.

It is true also that the following conditions are equivalent for a ring R:

1. R is a right coherent ring.
2. The module R_R is a coherent module.
3. Every finitely presented right R module is coherent.

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.

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