In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a finitely generated module is a
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Mo ...
that has a
finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marke ...
generating set
In mathematics and physics, the term generator or generating set may refer to any of a number of related concepts. The underlying concept in each case is that of a smaller set of objects, together with a set of operations that can be applied to ...
. A finitely generated module over a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
''R'' may also be called a finite ''R''-module, finite over ''R'', 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
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noether ...
the concepts of finitely generated, finitely presented and coherent modules coincide.
A finitely generated module over a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
is simply a
finite-dimensional
In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to disti ...
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
, and a finitely generated module over the
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s is simply a
finitely generated abelian group
In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x_1,\dots,x_s in G such that every x in G can be written in the form x = n_1x_1 + n_2x_2 + \cdots + n_sx_s for some integers n_1,\dots, n ...
.
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
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
is referred to as a
generating set
In mathematics and physics, the term generator or generating set may refer to any of a number of related concepts. The underlying concept in each case is that of a smaller set of objects, together with a set of operations that can be applied to ...
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:
:
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 there is a finite generating set that is included in ''S'', since only finitely many elements in ''S'' are needed to express 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
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
. For example the set of the
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s is a generating set of
viewed as
-module, and a generating set formed from prime numbers has at least two elements, while the
singleton
Singleton may refer to:
Sciences, technology Mathematics
* Singleton (mathematics), a set with exactly one element
* Singleton field, used in conformal field theory Computing
* Singleton pattern, a design pattern that allows only one instance ...
is also a generating set.
In the case where the
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Mo ...
''M'' is a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
over a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
''R'', and the generating set is
linearly independent
In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
, ''n'' is ''well-defined'' and is referred to as the
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
of ''M'' (''well-defined'' means that any
linearly independent
In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
generating set has ''n'' elements: this is the
dimension theorem for vector spaces
In mathematics, the dimension theorem for vector spaces states that all bases of a vector space have equally many elements. This number of elements may be finite or infinite (in the latter case, it is a cardinal number), and defines the dimension ...
).
Any module is the union of the
directed set
In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation \,\leq\, (that is, a preorder), with the additional property that every pair of elements has ...
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
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals c ...
s. If any increasing chain of submodules stabilizes (i.e., any submodule is finitely generated), then the module ''M'' is called a
Noetherian module In abstract algebra, a Noetherian module is a module that satisfies the ascending chain condition on its submodules, where the submodules are partially ordered by inclusion.
Historically, Hilbert was the first mathematician to work with the proper ...
.
Examples
* If a module is generated by one element, it is called a
cyclic module In mathematics, more specifically in ring theory, a cyclic module or monogenous module is a module over a ring that is generated by one element. The concept is a generalization of the notion of a cyclic group, that is, an Abelian group (i.e. Z-mod ...
.
* 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
In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral doma ...
: 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
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s Z coincide with the
finitely generated abelian group
In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x_1,\dots,x_s in G such that every x in G can be written in the form x = n_1x_1 + n_2x_2 + \cdots + n_sx_s for some integers n_1,\dots, n ...
s. These are completely classified by the
structure theorem, taking Z as the principal ideal domain.
* Finitely generated (say left) modules over a
division ring
In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element us ...
are precisely finite dimensional vector spaces (over the division ring).
Some facts
Every
homomorphic image of a finitely generated module is finitely generated. In general,
submodule
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the mod ...
s of finitely generated modules need not be finitely generated. As an example, consider the ring ''R'' = Z
1, ''X''2, ...">'X''1, ''X''2, ...of all
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
s in
countably many
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural number ...
variables. ''R'' itself is a finitely generated ''R''-module (with 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 In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite lengt ...
if every submodule is finitely generated. A finitely generated module over a
Noetherian ring
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noether ...
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 In mathematics, a finitely generated algebra (also called an algebra of finite type) is a commutative associative algebra ''A'' over a field ''K'' where there exists a finite set of elements ''a''1,...,''a'n'' of ''A'' such that every element of ...
. Conversely, if a finitely generated algebra is integral (over the coefficient ring), then it is finitely generated module. (See
integral element In commutative algebra, an element ''b'' of a commutative ring ''B'' is said to be integral over ''A'', a subring of ''B'', if there are ''n'' ≥ 1 and ''a'j'' in ''A'' such that
:b^n + a_ b^ + \cdots + a_1 b + a_0 = 0.
That is to say, ''b'' is ...
for more.)
Let 0 → ''M''′ → ''M'' → ''M''′′ → 0 be an
exact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the context o ...
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 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
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
''R''-endomorphism of a finitely generated module ''M'', then ''f'' is also
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
, and hence is an
automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
of ''M''. This says simply that ''M'' is a
Hopfian module. Similarly, an
Artinian module
In mathematics, specifically abstract algebra, an Artinian module is a module that satisfies the descending chain condition on its poset of submodules. They are for modules what Artinian rings are for rings, and a ring is Artinian if and only if it ...
''M'' is
coHopfian: any injective endomorphism ''f'' is also a surjective endomorphism.
Any ''R''-module is an
inductive limit
In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any catego ...
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
In mathematics, the Tor functors are the derived functors of the tensor product of modules over a ring. Along with the Ext functor, Tor is one of the central concepts of homological algebra, in which ideas from algebraic topology are used to con ...
.)
An example of a link between finite generation and
integral element In commutative algebra, an element ''b'' of a commutative ring ''B'' is said to be integral over ''A'', a subring of ''B'', if there are ''n'' ≥ 1 and ''a'j'' in ''A'' such that
:b^n + a_ b^ + \cdots + a_1 b + a_0 = 0.
That is to say, ''b'' is ...
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 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
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) ...
''R''
'x''is finitely generated by 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:
* ''A'' is a finitely generated ''R'' module.
* ''A'' is both a finitely generated ring over ''R'' and an
integral extension In commutative algebra, an element ''b'' of a commutative ring ''B'' is said to be integral over ''A'', a subring of ''B'', if there are ''n'' ≥ 1 and ''a'j'' in ''A'' such that
:b^n + a_ b^ + \cdots + a_1 b + a_0 = 0.
That is to say, ''b'' i ...
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
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
In mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group ''A'' is the cardinality of a maximal linearly independent subset. The rank of ''A'' determines the size of the largest free abelian group contained in ''A''. If ''A'' ...
''). Since
,
is a
torsion module
In mathematics, specifically in ring theory, a torsion element is an element of a module that yields zero when multiplied by some non-zero-divisor of the ring. The torsion submodule of a module is the submodule formed by the torsion elements. ...
. When ''A'' is Noetherian, by
generic freeness In algebraic geometry and commutative algebra, the theorems of generic flatness and generic freeness state that under certain hypotheses, a sheaf (mathematics), sheaf of module (mathematics), modules on a scheme (mathematics), scheme is flat morphis ...
, there is an element ''f'' (depending on ''M'') such that