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| In [[mathematics]], a '''finitely generated module''' is a [[module (mathematics)|module]] that has a finite [[generating set]]. A finitely generated ''R''-module also may be called a '''finite ''R''-module''' or '''finite over ''R'''''.<ref>For example, Matsumura uses this terminology.</ref>
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| 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 related, finitely presented and coherent modules all coincide.
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| 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]].
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| ==Formal definition==
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| The left ''R''-module ''M'' is finitely generated [[if and only if]] there exist ''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''n''</sub> in ''M'' such that for all ''x'' in ''M'', there exist ''r''<sub>1</sub>, ''r''<sub>2</sub>, ..., ''r''<sub>''n''</sub> in ''R'' with ''x'' = ''r''<sub>1</sub>''a''<sub>1</sub> + ''r''<sub>2</sub>''a''<sub>2</sub> + ... + ''r''<sub>''n''</sub>''a''<sub>''n''</sub>.
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| The set {''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''n''</sub>} is referred to as a '''generating set''' for ''M'' in this case.
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| 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]]).
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| == Examples ==
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| * Let ''R'' be an integral domain with ''K'' its field of fractions. Then every ''R''-submodule of ''K'' is a [[fractional ideal]]. If ''R'' is Noetherian, every fractional ideal arises in this way.
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| * 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.
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| * Finitely generated modules over [[division ring]]s{{Citation needed||date=July 2013}} are precisely finite dimensional vector spaces.
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| ==Some facts==
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| 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''<sub>1</sub>, ''X''<sub>2</sub>, ...] 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.
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| 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 algebra over a Noetherian ring is again a Noetherian ring.
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| 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.)
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| 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).
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| 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 ''B'' ⊗<sub>''A''</sub> ''F'' is finitely generated (resp. finitely presented).{{sfn|Bourbaki|1998|loc=Ch 1, §3, no. 6, Proposition 11}}
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| == Finitely generated modules over a commutative ring ==
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| 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}}
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| 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]].)
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| 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 ''G'' = {''x''<sub>1</sub>, ..., ''x''<sub>n</sub>} 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''[''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:{{sfn|Kaplansky|1970|loc=Theorem 17|p=11}}
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| * ''A'' is a finitely generated ''R'' module.
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| * ''A'' is both a finitely generated ring over ''R'' and an [[integral element|integral extension]] of ''R''.
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| ==Equivalent definitions and finitely cogenerated modules==
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| The following conditions are equivalent to ''M'' being finitely generated (f.g.):
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| *For any family of submodules {''N<sub>i</sub>'' | ''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''.
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| *For any [[Total order#Chains|chain]] of submodules {''N<sub>i</sub>'' | ''i'' ∈ I} in ''M'', if <math>\bigcup_{i\in I}N_i=M\,</math>, then ''N<sub>i</sub>'' = ''M'' for some ''i'' in ''I''.
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| *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''.
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| 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.):
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| *For any family of submodules {''N<sub>i</sub>'' | ''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''.
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| *For any chain of submodules {''N<sub>i</sub>'' | ''i'' ∈ I} in ''M'', if <math>\bigcap_{i\in I}N_i=\{0\}\,</math>, then ''N<sub>i</sub>'' = {0} for some ''i'' in ''I''.
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| *If <math>\phi:M\to \prod_{i\in I}R\,</math> is a [[monomorphism]], then <math>\phi:M\to \prod_{i\in F}R\,</math> is a monomorphism for some finite subset ''F'' of ''I''.
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| 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'':
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| * ''M'' is Noetherian if and only if every submodule of ''N'' of ''M'' is f.g.
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| * ''M'' is Artinian if and only if every quotient module ''M''/''N'' is f.cog.
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| * ''M'' is f.g. if and only if ''J''(''M'') is a [[superfluous submodule]] of ''M'', and ''M''/''J''(''M'') is f.g.
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| * ''M'' is f.cog. if and only if soc(''M'') is an [[essential submodule]] of ''M'', and soc(''M'') is f.g.
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| * If ''M'' is a [[semisimple module]] (such as soc(''N'') for any module ''N''), it is f.g. if and only if f.cog.
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| * If ''M'' is f.g. and nonzero, then ''M'' has a [[maximal submodule]] and any quotient module ''M''/''N'' is f.g.
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| * If ''M'' is f.cog. and nonzero, then ''M'' has a minimal submodule, and any submodule ''N'' of ''M'' is f.cog.
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| * If ''N'' and ''M''/''N'' are f.g. then so is ''M''. The same is true if "f.g." is replaced with "f.cog."
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| 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.
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| ==<span id="Finitely presented module"><span>Finitely presented, finitely related, and coherent modules==
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| Another formulation is this: a finitely generated module ''M'' is one for which there is an [[epimorphism]]
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| :f : ''R<sup>k</sup>'' → ''M''.
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| Suppose now there is an epimorphism,
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| :φ : ''F'' → ''M''.
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| for a module ''M'' and free module ''F''.
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| * 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)).
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| * If the kernel of φ is finitely generated and ''F'' has finite rank (i.e. ''F''=''R''<sup>''k''</sup>), 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<sup>k</sup>'') and finitely many relations (the generators of ker(φ)).
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| *A '''coherent module''' ''M'' is a finitely generated module whose finitely generated submodules are finitely presented.
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| 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'', all four conditions are actually equivalent.
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| Some crossover occurs for projective or flat modules. A finitely generated projective module is finitely presented, and a finitely related flat module is projective.
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| It is true also that the following conditions are equivalent for a ring ''R'':
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| # ''R'' is a right [[coherent ring]].
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| # The module ''R''<sub>''R''</sub> is a coherent module.
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| # Every finitely presented right ''R'' module is coherent.
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| 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.
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| == See also ==
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| *[[Integral element]]
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| *[[Artin–Rees lemma]]
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| *[[Countably generated module]]
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| ==References==
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| {{reflist}}
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| ==Textbooks==
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| *{{citation
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| |author1=Atiyah, M. F.
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| |author2=Macdonald, I. G.
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| |title=Introduction to commutative algebra
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| |publisher=Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont.
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| |year=1969
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| |pages=ix+128
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| |mr=0242802 (39 #4129)
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| }}
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| * [[Nicolas Bourbaki|Bourbaki, Nicolas]], ''Commutative algebra. Chapters 1--7''. Translated from the French. Reprint of the 1989 English translation. Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 1998. xxiv+625 pp. ISBN 3-540-64239-0
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| *{{citation |author=Kaplansky, Irving |title=Commutative rings |publisher=Allyn and Bacon Inc. |place=Boston, Mass. |year=1970 |pages=x+180 |mr=0254021 }}
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| *{{Citation | last1=Lam | first1=T. Y. | title=Lectures on modules and rings | publisher=Springer-Verlag | series=Graduate Texts in Mathematics No. 189 | isbn=978-0-387-98428-5 | year=1999}}
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| *{{Citation | last1=Lang | first1=Serge | author1-link=Serge Lang | title=Algebra | publisher=[[Addison-Wesley]] | edition=3rd | isbn=978-0-201-55540-0 | year=1997}}
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| *{{citation
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| |author=Matsumura, Hideyuki
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| |title=Commutative ring theory
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| |series=Cambridge Studies in Advanced Mathematics
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| |volume=8
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| |edition=2
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| |note=Translated from the Japanese by M. Reid
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| |publisher=Cambridge University Press
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| |place=Cambridge
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| |year=1989
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| |pages=xiv+320
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| |isbn=0-521-36764-6
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| |mr=1011461 (90i:13001)
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| }}
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| [[Category:Module theory]]
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| [[fr:Module sur un anneau#Propriétés de finitude]]
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