imported>1234qwer1234qwer4: /* Generic rank */ clarify

2025-05-06T01:09:11Z

Generic rank: clarify

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{{Short description|In algebra, module with a finite generating set}}
In [[mathematics]], a '''finitely generated module''' is a [[module (mathematics)|module]] that has a [[Finite set|finite]] [[Generating set of a module|generating set]]. A finitely generated module over a [[Ring (mathematics)|ring]] ''R'' may also be called a '''finite ''R''-module''', '''finite over ''R''''',<ref>For example, Matsumura uses this terminology.</ref> 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]] the concepts of finitely generated, finitely presented and coherent modules coincide.

A finitely generated module over a [[Field (mathematics)|field]] is simply a [[Dimension (vector space)|finite-dimensional]] [[vector space]], and a finitely generated module over the [[integer]]s is simply a [[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 (mathematics)|set]] {''a''1, ''a''2, ..., ''a''''n''} is referred to as a [[generating set of a module|generating set]] 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 [[module homomorphism|''R''-linear map]]:
:<math>R^n \to M</math>
for some ''n''; in other words, ''M'' is a [[Quotient module|quotient]] of a [[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]]. For example the set of the [[prime number]]s is a generating set of <math>\mathbb Z</math> viewed as <math>\mathbb Z</math>-module, and a generating set formed from prime numbers has at least two elements, while the [[singleton (mathematics)|singleton]]{{math|{{mset|1}}}} is also a generating set.

In the case where the [[module (mathematics)|module]] ''M'' is a [[vector space]] over a [[field (mathematics)|field]] ''R'', and the generating set is [[linearly independent]], ''n'' is ''well-defined'' and is referred to as the [[dimension of a vector space|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''. This fact with [[Zorn's lemma]] implies that every nonzero finitely generated module admits [[maximal submodule]]s. If any increasing chain of submodules stabilizes (i.e., any submodule is finitely generated), then the module ''M'' is called a [[Noetherian module]].

== Examples ==
* If a module is generated by one element, it is called a [[cyclic module]].
* Let ''R'' be an [[integral domain]] with ''K'' its [[field of fractions]]. Then every finitely generated ''R''-submodule ''I'' of ''K'' is a [[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 [[integer]]s '''Z''' coincide with the [[finitely generated abelian group]]s. These are completely classified by the [[Structure theorem for finitely generated modules over a principal ideal domain|structure theorem]], taking '''Z''' as the principal ideal domain.
* Finitely generated (say left) modules over a [[division ring]] are precisely finite dimensional vector spaces (over the division ring).

==Some facts==

Every [[module homomorphism|homomorphic image]] of a finitely generated module is finitely generated. In general, [[submodule]]s of finitely generated modules need not be finitely generated. As an example, consider the ring ''R'' = '''Z'''[''X''1, ''X''2, ...] of all [[polynomial]]s in [[countable|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 module|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 module|faithfully flat]] right ''A''-module. Then a left ''A''-module ''F'' is finitely generated (resp. finitely presented) if and only if the ''B''-module {{nowrap|''B'' ⊗''A'' ''F''}} is finitely generated (resp. finitely presented).{{sfn|Bourbaki|1998|loc=Ch 1, §3, no. 6, Proposition 11}}

== 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''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]] ''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''.

== Generic rank ==
Let ''M'' be a finitely generated module over an integral domain ''A'' with the field of fractions ''K''. Then the dimension <math>\operatorname{dim}_K (M \otimes_A K)</math> 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 <math>(M/F)_{(0)} = M_{(0)}/F_{(0)} = 0</math>, <math>M/F</math> is a [[torsion module]]. When ''A'' is Noetherian, by [[generic freeness]], there is an element ''f'' (depending on ''M'') such that <math>M[f^{-1}]</math> is a free <math>A[f^{-1}]</math>-module. Then the rank of this free module is the generic rank of ''M''.

Now suppose the integral domain ''A'' is an <math>\mathbb{N}</math>-[[graded algebra]] over a field ''k'' generated by finitely many homogeneous elements of degrees <math>d_i</math>. Suppose ''M'' is graded as well and let <math>P_M(t) = \sum (\operatorname{dim}_k M_n) t^n</math> be the [[Poincaré series (modular form)|Poincaré series]] of ''M''.
By the [[Hilbert–Serre theorem]], there is a polynomial ''F'' such that <math>P_M(t) = F(t) \prod (1-t^{d_i})^{-1}</math>. Then <math>F(1)</math> is the generic rank of ''M''.<ref>{{harvnb|Springer|1977|loc=Theorem 2.5.6.}}</ref>

A finitely generated module over a [[principal ideal domain]] is [[torsion-free module|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 <math>f M \subset F</math>. Then <math>fM</math> is free since it is a submodule of a free module and ''A'' is a PID. But now <math>f: M \to fM</math> 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 module|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.):
*For any family of submodules {''Ni'' | ''i'' ∈ ''I''} in ''M'', if <math>\sum_{i\in I}N_i=M\,</math>, then <math>\sum_{i\in F}N_i=M\,</math> for some finite [[subset]] ''F'' of ''I''.
*For any [[Total order#Chains|chain]] of submodules {''Ni'' | ''i'' ∈ ''I''} in ''M'', if <math>\bigcup_{i\in I}N_i=M\,</math>, then {{nowrap|1=''Ni'' = ''M''}} for some ''i'' in ''I''.
*If <math>\phi:\bigoplus_{i\in I}R\to M\,</math> is an [[epimorphism]], then the restriction <math>\phi:\bigoplus_{i\in F}R\to M\,</math> 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]]. The conditions are also convenient to define a [[duality (mathematics)|dual]] 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 <math>\bigcap_{i\in I}N_i=\{0\}\,</math>, then <math>\bigcap_{i\in F}N_i=\{0\}\,</math> for some finite subset ''F'' of ''I''.
*For any chain of submodules {''Ni'' | ''i'' ∈ ''I''} in ''M'', if <math>\bigcap_{i\in I}N_i=\{0\}\,</math>, then ''Ni'' = {{mset|0}} for some ''i'' in ''I''.
*If <math>\phi:M\to \prod_{i\in I}N_i\,</math> is a [[monomorphism]], where each <math>N_i</math> is an ''R'' module, then <math>\phi:M\to \prod_{i\in F}N_i\,</math> 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]] ''J''(''M'') and [[socle (mathematics)|socle]] 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]] of ''M'', and ''M''/''J''(''M'') is f.g.
* ''M'' is f.cog. if and only if soc(''M'') is an [[essential submodule]] of ''M'', and soc(''M'') is f.g.
* If ''M'' is a [[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]] 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]]. 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 [[uniform module#Hollow modules and co-uniform dimension|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]] 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 (algebra)|kernel]] 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. {{nowrap|1=''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 {{nowrap|1=''F'' = ''R''''k''}}) and finitely many relations (the generators of ker(''φ'')). See also: [[free presentation]]. Finitely presented modules can be characterized by an abstract property within the [[category of modules|category of ''R''-modules]]: they are precisely the [[compact object (mathematics)|compact objects]] 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]] ''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'':
# ''R'' is a right [[coherent ring]].
# The module ''R''''R'' is a coherent module.
# 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 (mathematics)|category]] of coherent modules is an [[abelian category]], while, in general, neither finitely generated nor finitely presented modules form an abelian category.

== See also ==
*[[Integral element]]
*[[Artin–Rees lemma]]
*[[Countably generated module]]
*[[Finite algebra]]
*[[Coherent sheaf]], a generalization used in algebraic geometry

==References==
{{reflist}}

==Textbooks==
*{{citation
|last1=Atiyah |first1=M. F. |author-link1=Michael Atiyah
|last2=Macdonald |first2=I. G. |author-link2=Ian G. Macdonald
|title=Introduction to commutative algebra
|publisher=Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont.
|year=1969
|pages=ix+128
|mr=0242802
}}
* {{citation |author-link=Nicolas Bourbaki |last=Bourbaki |first=Nicolas |title=Commutative algebra. Chapters 1--7 Translated from the French. Reprint of the 1989 English translation |series=Elements of Mathematics |location=Berlin |publisher=Springer-Verlag |date=1998 |isbn=3-540-64239-0}}
*{{citation |last=Kaplansky |first=Irving |author-link=Irving Kaplansky |title=Commutative rings |publisher=Allyn and Bacon Inc. |place=Boston, Mass. |year=1970 |pages=x+180 |mr=0254021 }}
*{{Citation | last1=Lam | first1=T. Y. | author-link1=Tsit Yuen Lam | title=Lectures on modules and rings | publisher=Springer-Verlag | series=Graduate Texts in Mathematics No. 189 | isbn=978-0-387-98428-5 | year=1999}}
*{{Citation | last1=Lang | first1=Serge | author1-link=Serge Lang | title=Algebra | publisher=[[Addison-Wesley]] | edition=3rd | isbn=978-0-201-55540-0 | year=1997}}
*{{citation
|last=Matsumura |first=Hideyuki |author-link=Hideyuki Matsumura
|title=Commutative ring theory
|series=Cambridge Studies in Advanced Mathematics
|volume=8
|edition=2
|others=Translated from the Japanese by M. Reid
|publisher=Cambridge University Press
|place=Cambridge
|year=1989
|pages=xiv+320
|isbn=0-521-36764-6
|mr=1011461
}}
* {{Citation
| last=Springer
| first=Tonny A.
| title=Invariant theory
| series=Lecture Notes in Mathematics
| volume=585
| publisher=Springer
| year=1977
| doi=10.1007/BFb0095644
| isbn=978-3-540-08242-2
}}.

[[Category:Module theory]]

[[fr:Module sur un anneau#Propriétés de finitude]]

Finitely generated module - Revision history

imported>1234qwer1234qwer4: /* Generic rank */ clarify