# Finitely generated module

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In algebra, module with a finite generating set

In [mathematics](/source/Mathematics), a **finitely generated module** is a [module](/source/Module_(mathematics)) that has a [finite](/source/Finite_set) [generating set](/source/Generating_set_of_a_module). A finitely generated module over a [ring](/source/Ring_(mathematics)) *R* may also be called a **finite *R*-module**, **finite over *R***,[1] or a **module of finite type**.

Related concepts include **finitely cogenerated modules**, **finitely presented modules**, **finitely related modules** and **coherent modules** all of which are defined below. Over a [Noetherian ring](/source/Noetherian_ring) the concepts of finitely generated, finitely presented and coherent modules coincide.

A finitely generated module over a [field](/source/Field_(mathematics)) is simply a [finite-dimensional](/source/Dimension_(vector_space)) [vector space](/source/Vector_space), and a finitely generated module over the [integers](/source/Integer) is simply a [finitely generated abelian group](/source/Finitely_generated_abelian_group).

## Definition

The left *R*-module *M* is finitely generated if there exist *a*1, *a*2, ..., *a**n* in *M* such that for any *x* in *M*, there exist *r*1, *r*2, ..., *r**n* in *R* with *x* = *r*1*a*1 + *r*2*a*2 + ... + *r**n**a**n*.

The [set](/source/Set_(mathematics)) {*a*1, *a*2, ..., *a**n*} is referred to as a [generating set](/source/Generating_set_of_a_module) of *M* in this case. 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](/source/Module_homomorphism):

- R n → M {\displaystyle R^{n}\to M}

for some *n*; in other words, *M* is a [quotient](/source/Quotient_module) of a [free module](/source/Free_module) of finite rank.

If a set *S* generates a module that is finitely generated, then there is a finite generating set that is included in *S*, since only finitely many elements in *S* are needed to express the generators in any finite generating set, and these finitely many elements form a generating set. However, it may occur that *S* does not contain any finite generating set of minimal [cardinality](/source/Cardinality). For example the set of the [prime numbers](/source/Prime_number) is a generating set of Z {\displaystyle \mathbb {Z} } viewed as Z {\displaystyle \mathbb {Z} } -module, and a generating set formed from prime numbers has at least two elements, while the [singleton](/source/Singleton_(mathematics)){1} is also a generating set.

In the case where the [module](/source/Module_(mathematics)) *M* is a [vector space](/source/Vector_space) over a [field](/source/Field_(mathematics)) *R*, and the generating set is [linearly independent](/source/Linearly_independent), *n* is *well-defined* and is referred to as the [dimension](/source/Dimension_of_a_vector_space) of *M* (*well-defined* means that any [linearly independent](/source/Linearly_independent) generating set has *n* elements: this is the [dimension theorem for vector spaces](/source/Dimension_theorem_for_vector_spaces)).

Any module is the union of the [directed set](/source/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*. This fact with [Zorn's lemma](/source/Zorn's_lemma) implies that every nonzero finitely generated module admits [maximal submodules](/source/Maximal_submodule). If any increasing chain of submodules stabilizes (i.e., any submodule is finitely generated), then the module *M* is called a [Noetherian module](/source/Noetherian_module).

## Examples

- If a module is generated by one element, it is called a [cyclic module](/source/Cyclic_module).

- Let *R* be an [integral domain](/source/Integral_domain) with *K* its [field of fractions](/source/Field_of_fractions). Then every finitely generated *R*-submodule *I* of *K* is a [fractional ideal](/source/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 [integers](/source/Integer) **Z** coincide with the [finitely generated abelian groups](/source/Finitely_generated_abelian_group). These are completely classified by the [structure theorem](/source/Structure_theorem_for_finitely_generated_modules_over_a_principal_ideal_domain), taking **Z** as the principal ideal domain.

- Finitely generated (say left) modules over a [division ring](/source/Division_ring) are precisely finite dimensional vector spaces (over the division ring).

## Some facts

Every [homomorphic image](/source/Module_homomorphism) of a finitely generated module is finitely generated. In general, [submodules](/source/Submodule) of finitely generated modules need not be finitely generated. As an example, consider the ring *R* = **Z**[*X*1, *X*2, ...] of all [polynomials](/source/Polynomial) in [countably many](/source/Countable) 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](/source/Noetherian_module) if every submodule is finitely generated. A finitely generated module over a [Noetherian ring](/source/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](/source/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](/source/Finitely_generated_algebra). Conversely, if a finitely generated algebra is integral (over the coefficient ring), then it is finitely generated module. (See [integral element](/source/Integral_element) for more.)

Let 0 → *M*′ → *M* → *M*′′ → 0 be an [exact sequence](/source/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](/source/Faithfully_flat_module) 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).[2]

## Finitely generated modules over a commutative ring

For finitely generated modules over a commutative ring *R*, [Nakayama's lemma](/source/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](/source/Surjective) *R*-endomorphism of a finitely generated module *M*, then *f* is also [injective](/source/Injective_function), and hence is an [automorphism](/source/Automorphism) of *M*.[3] This says simply that *M* is a [Hopfian module](/source/Hopfian_module). Similarly, an [Artinian module](/source/Artinian_module) *M* is [coHopfian](/source/Hopfian_object): any injective endomorphism *f* is also a surjective endomorphism.[4] The [Forster–Swan theorem](/source/Forster%E2%80%93Swan_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](/source/Inductive_limit) of finitely generated *R*-submodules. This is useful for weakening an assumption to the finite case (e.g., the [characterization of flatness](/source/Flat_module#Homological_algebra) with the [Tor functor](/source/Tor_functor)).

An example of a link between finite generation and [integral elements](/source/Integral_element) 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* = {*x*1, ..., *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](/source/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:[5]

- *A* is a finitely generated *R* module.

- *A* is both a finitely generated ring over *R* and an [integral extension](/source/Integral_element) of *R*.

## 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 ) {\displaystyle \operatorname {dim} _{K}(M\otimes _{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](/source/Rank_of_an_abelian_group)*). Since ( M / F ) ( 0 ) = M ( 0 ) / F ( 0 ) = 0 {\displaystyle (M/F)_{(0)}=M_{(0)}/F_{(0)}=0} , M / F {\displaystyle M/F} is a [torsion module](/source/Torsion_module). When *A* is Noetherian, by [generic freeness](/source/Generic_freeness), there is an element *f* (depending on *M*) such that M [ f − 1 ] {\displaystyle M[f^{-1}]} is a free A [ f − 1 ] {\displaystyle A[f^{-1}]} -module. Then the rank of this free module is the generic rank of *M*.

Now suppose the integral domain *A* is an N {\displaystyle \mathbb {N} } -[graded algebra](/source/Graded_algebra) over a field *k* generated by finitely many homogeneous elements of degrees d i {\displaystyle d_{i}} . Suppose *M* is graded as well and let P M ( t ) = ∑ ( dim k ⁡ M n ) t n {\displaystyle P_{M}(t)=\sum (\operatorname {dim} _{k}M_{n})t^{n}} be the [Poincaré series](/source/Poincar%C3%A9_series_(modular_form)) of *M*. By the [Hilbert–Serre theorem](/source/Hilbert%E2%80%93Serre_theorem), there is a polynomial *F* such that P M ( t ) = F ( t ) ∏ ( 1 − t d i ) − 1 {\displaystyle P_{M}(t)=F(t)\prod (1-t^{d_{i}})^{-1}} . Then F ( 1 ) {\displaystyle F(1)} is the generic rank of *M*.[6]

A finitely generated module over a [principal ideal domain](/source/Principal_ideal_domain) is [torsion-free](/source/Torsion-free_module) if and only if it is free. This is a consequence of the [structure theorem for finitely generated modules over a principal ideal domain](/source/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 {\displaystyle fM\subset F} . Then f M {\displaystyle fM} is free since it is a submodule of a free module and *A* is a PID. But now f : M → f M {\displaystyle f:M\to fM} is an isomorphism since *M* is torsion-free.

By the same argument as above, a finitely generated module over a [Dedekind domain](/source/Dedekind_domain) *A* (or more generally a [semi-hereditary ring](/source/Semi-hereditary_ring)) is torsion-free if and only if it is [projective](/source/Projective_module); 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.):

- For any family of submodules {*Ni* | *i* ∈ *I*} in *M*, if ∑ i ∈ I N i = M {\displaystyle \sum _{i\in I}N_{i}=M\,} , then ∑ i ∈ F N i = M {\displaystyle \sum _{i\in F}N_{i}=M\,} for some finite [subset](/source/Subset) *F* of *I*.

- For any [chain](/source/Total_order#Chains) of submodules {*Ni* | *i* ∈ *I*} in *M*, if ⋃ i ∈ I N i = M {\displaystyle \bigcup _{i\in I}N_{i}=M\,} , then *Ni* = *M* for some *i* in *I*.

- If ϕ : ⨁ i ∈ I R → M {\displaystyle \phi :\bigoplus _{i\in I}R\to M\,} is an [epimorphism](/source/Epimorphism), then the restriction ϕ : ⨁ i ∈ F R → M {\displaystyle \phi :\bigoplus _{i\in F}R\to M\,} is an epimorphism for some finite subset *F* of *I*.

From these conditions it is easy to see that being finitely generated is a property preserved by [Morita equivalence](/source/Morita_equivalence). The conditions are also convenient to define a [dual](/source/Duality_(mathematics)) notion of a **finitely cogenerated module** *M*. The following conditions are equivalent to a module being finitely cogenerated (f.cog.):

- For any family of submodules {*Ni* | *i* ∈ *I*} in *M*, if ⋂ i ∈ I N i = { 0 } {\displaystyle \bigcap _{i\in I}N_{i}=\{0\}\,} , then ⋂ i ∈ F N i = { 0 } {\displaystyle \bigcap _{i\in F}N_{i}=\{0\}\,} for some finite subset *F* of *I*.

- For any chain of submodules {*Ni* | *i* ∈ *I*} in *M*, if ⋂ i ∈ I N i = { 0 } {\displaystyle \bigcap _{i\in I}N_{i}=\{0\}\,} , then *Ni* = {0} for some *i* in *I*.

- If ϕ : M → ∏ i ∈ I N i {\displaystyle \phi :M\to \prod _{i\in I}N_{i}\,} is a [monomorphism](/source/Monomorphism), where each N i {\displaystyle N_{i}} is an *R* module, then ϕ : M → ∏ i ∈ F N i {\displaystyle \phi :M\to \prod _{i\in F}N_{i}\,} is a monomorphism for some finite subset *F* of *I*.

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

- *M* is Noetherian if and only if every submodule *N* of *M* is f.g.

- *M* is Artinian if and only if every quotient module *M*/*N* is f.cog.

- *M* is f.g. if and only if *J*(*M*) is a [superfluous submodule](/source/Superfluous_submodule) of *M*, and *M*/*J*(*M*) is f.g.

- *M* is f.cog. if and only if soc(*M*) is an [essential submodule](/source/Essential_submodule) of *M*, and soc(*M*) is f.g.

- If *M* is a [semisimple module](/source/Semisimple_module) (such as soc(*N*) for any module *N*), it is f.g. if and only if f.cog.

- If *M* is f.g. and nonzero, then *M* has a [maximal submodule](/source/Maximal_submodule) and any quotient module *M*/*N* is f.g.

- If *M* is f.cog. and nonzero, then *M* has a minimal submodule, and any submodule *N* of *M* is f.cog.

- If *N* and *M*/*N* are f.g. then so is *M*. The same is true if "f.g." is replaced with "f.cog."

Finitely cogenerated modules must have finite [uniform dimension](/source/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](/source/Uniform_module#Hollow_modules_and_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](/source/Epimorphism) mapping *Rk* onto *M* :

- f : *Rk* → *M*.

Suppose now there is an epimorphism,

- *φ* : *F* → *M*.

for a module *M* and free module *F*.

- If the [kernel](/source/Kernel_(algebra)) of *φ* is finitely generated, then *M* is called a **finitely related module**. Since *M* is isomorphic to *F*/ker(*φ*), this basically expresses that *M* is obtained by taking a free module and introducing finitely many relations within *F* (the generators of ker(*φ*)).

- If the kernel of *φ* is finitely generated and *F* has finite rank (i.e. *F* = *R**k*), then *M* is said to be a **finitely presented module**. Here, *M* is specified using finitely many generators (the images of the *k* generators of *F* = *R**k*) and finitely many relations (the generators of ker(*φ*)). See also: [free presentation](/source/Free_presentation). Finitely presented modules can be characterized by an abstract property within the [category of *R*-modules](/source/Category_of_modules): they are precisely the [compact objects](/source/Compact_object_(mathematics)) in this category.

- A **coherent module** *M* is a finitely generated module whose finitely generated submodules are finitely presented.

Over any ring *R*, coherent modules are finitely presented, and finitely presented modules are both finitely generated and finitely related. For a [Noetherian ring](/source/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](/source/Coherent_ring).

1. The module *R**R* is a coherent module.

1. 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](/source/Category_(mathematics)) of coherent modules is an [abelian category](/source/Abelian_category), while, in general, neither finitely generated nor finitely presented modules form an abelian category.

## See also

- [Integral element](/source/Integral_element)

- [Artin–Rees lemma](/source/Artin%E2%80%93Rees_lemma)

- [Countably generated module](/source/Countably_generated_module)

- [Finite algebra](/source/Finite_algebra)

- [Coherent sheaf](/source/Coherent_sheaf), a generalization used in algebraic geometry

## References

1. **[^](#cite_ref-1)** For example, Matsumura uses this terminology.

1. **[^](#cite_ref-FOOTNOTEBourbaki1998Ch_1,_§3,_no._6,_Proposition_11_2-0)** [Bourbaki 1998](#CITEREFBourbaki1998), Ch 1, §3, no. 6, Proposition 11.

1. **[^](#cite_ref-FOOTNOTEMatsumura1989Theorem_2.4_3-0)** [Matsumura 1989](#CITEREFMatsumura1989), Theorem 2.4.

1. **[^](#cite_ref-FOOTNOTEAtiyahMacdonald1969Exercise_6.1_4-0)** [Atiyah & Macdonald 1969](#CITEREFAtiyahMacdonald1969), Exercise 6.1.

1. **[^](#cite_ref-FOOTNOTEKaplansky197011Theorem_17_5-0)** [Kaplansky 1970](#CITEREFKaplansky1970), p. 11, Theorem 17.

1. **[^](#cite_ref-6)** [Springer 1977](#CITEREFSpringer1977), Theorem 2.5.6.

## Textbooks

- [Atiyah, M. F.](/source/Michael_Atiyah); [Macdonald, I. G.](/source/Ian_G._Macdonald) (1969), *[Introduction to Commutative Algebra](/source/Introduction_to_Commutative_Algebra)*, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., pp. ix+128, [MR](/source/MR_(identifier)) [0242802](https://mathscinet.ams.org/mathscinet-getitem?mr=0242802)

- [Bourbaki, Nicolas](/source/Nicolas_Bourbaki) (1998), *Commutative algebra. Chapters 1--7 Translated from the French. Reprint of the 1989 English translation*, Elements of Mathematics, Berlin: Springer-Verlag, [ISBN](/source/ISBN_(identifier)) [3-540-64239-0](https://en.wikipedia.org/wiki/Special:BookSources/3-540-64239-0)

- [Kaplansky, Irving](/source/Irving_Kaplansky) (1970), *Commutative rings*, Boston, Mass.: Allyn and Bacon Inc., pp. x+180, [MR](/source/MR_(identifier)) [0254021](https://mathscinet.ams.org/mathscinet-getitem?mr=0254021)

- [Lam, T. Y.](/source/Tsit_Yuen_Lam) (1999), *Lectures on modules and rings*, Graduate Texts in Mathematics No. 189, Springer-Verlag, [ISBN](/source/ISBN_(identifier)) [978-0-387-98428-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-98428-5)

- [Lang, Serge](/source/Serge_Lang) (1997), *Algebra* (3rd ed.), [Addison-Wesley](/source/Addison-Wesley), [ISBN](/source/ISBN_(identifier)) [978-0-201-55540-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-201-55540-0)

- [Matsumura, Hideyuki](/source/Hideyuki_Matsumura) (1989), *Commutative ring theory*, Cambridge Studies in Advanced Mathematics, vol. 8, Translated from the Japanese by M. Reid (2 ed.), Cambridge: Cambridge University Press, pp. xiv+320, [ISBN](/source/ISBN_(identifier)) [0-521-36764-6](https://en.wikipedia.org/wiki/Special:BookSources/0-521-36764-6), [MR](/source/MR_(identifier)) [1011461](https://mathscinet.ams.org/mathscinet-getitem?mr=1011461)

- Springer, Tonny A. (1977), *Invariant theory*, Lecture Notes in Mathematics, vol. 585, Springer, [doi](/source/Doi_(identifier)):[10.1007/BFb0095644](https://doi.org/10.1007%2FBFb0095644), [ISBN](/source/ISBN_(identifier)) [978-3-540-08242-2](https://en.wikipedia.org/wiki/Special:BookSources/978-3-540-08242-2).

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Adapted from the Wikipedia article [Finitely generated module](https://en.wikipedia.org/wiki/Finitely_generated_module) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Finitely_generated_module?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.