Module
An Abelian group with the distributive action of a ring. A module is a generalization of a (linear) vector space over a field , when
is replaced by a ring.
Let a ring be given. An additive Abelian group
is called a left
-module if there is a mapping
whose value on a pair
, for
,
, written
, satisfies the axioms:
1) ;
2) ;
3) . If
is a ring with identity, then it is usual to require in addition that for any
,
. A module with this property is called unitary or unital (cf. Unitary module).
Right -modules are defined similarly; axiom 3) is replaced by
. Any right
-module can be considered as a left
-module over the opposite ring
anti-isomorphic to
; hence, corresponding to any result about right
-modules there is a result about left
-modules, and conversely. When
is commutative, any left
-module can be considered as a right
-module and the distinction between left and right modules disappears. Below only left
-modules are discussed.
The simplest examples of modules (finite Abelian groups; they are -modules) were known already to C.F. Gauss as class groups of binary quadratic forms. The general notion of a module was first encountered in the 1860's till 1880's in the work of R. Dedekind and L. Kronecker, devoted to the arithmetic of algebraic number and function fields. At approximately the same time research on finite-dimensional associative algebras, in particular, group algebras of finite groups (B. Pierce, F. Frobenius), led to the study of ideals of certain non-commutative rings. At first the theory of modules was developed primarily as a theory of ideals of a ring. Only later, in the work of E. Noether and W. Krull, it was observed that it was more convenient to formulate and prove many results in terms of arbitrary modules, and not just ideals. Subsequent developments of the theory of modules were connected with the application of methods and ideas of the theory of categories (cf. Category), in particular, methods of homological algebra.
Examples of modules.
1) Any Abelian group is a module over the ring of integers
. For
and
the product
is defined as the result of adding
to itself
times.
2) When is a field, the notion of a unitary
-module is exactly equivalent to the notion of a linear vector space over
.
3) An -dimensional vector space
over a field
(provided with coordinates) can be considered as a module over the ring
of all
-matrices with coefficients in
. For
and
the product
is defined as multiplication of the matrix
by the column of coordinates of the vector
.
4) An associative ring (cf. Associative rings and algebras) is a left
-module. Multiplication of elements of the ring by elements of the module is ordinary multiplication in
.
5) The set of differential forms on a smooth manifold has the natural structure of a module over the ring of all smooth functions on
.
6) Connected with any Abelian group is the associative ring with identity,
, of all endomorphisms of
. The group
has a natural
-module structure.
If there is an -module structure on
, for some ring
, then the mapping
is an endomorphism of
for any
. Associating with the element
the endomorphism of
that it generates, one obtains a homomorphism
of
into
. Conversely, any homomorphism
defines the structure of an
-module on
. Thus, the specification of an
-module structure on an Abelian group
is equivalent to the specification of a homomorphism of rings
. Such a homomorphism is also called a representation of the ring
, and
is called a representation module. Connected with any representation
is a two-sided ideal
, consisting of the
such that
for all
. This ideal is called the annihilator of the module
. When
, the representation is called faithful and
is called a faithful module (or faithful representation).
It is obvious that a module can also be considered as a module over the quotient ring
. In particular, although the definition of a module does not assume the associativity of
, the ring
is always associative. Therefore, in the majority of cases the discussion may be restricted to modules over associative rings. Everywhere below, unless stated otherwise,
is assumed to be associative.
-modules.
Let be a group. An additive Abelian group
is called a left
-module if there is a mapping
whose value at a pair
, where
,
, is written as
, and where for any
the mapping
is an endomorphism of
; for any
,
,
; and for all
,
, where 1 is the identity of
. For any
the mapping
is an automorphism of the group
.
Right -modules may be defined similarly.
Examples of
-modules.
1) Let be a Galois extension of a field
with Galois group
. Then the additive and multiplicative groups of
have the natural structure of
-modules. If
is an algebraic number field, then other
-modules are: the additive group of the ring of integers of
, the group of units of
, the group of divisors and the divisor class group of
, etc. A module over a Galois group is called a Galois module.
2) Let an extension of an Abelian group be given, that is, an exact sequence of groups
![]() |
where is an Abelian normal subgroup of
and
is an arbitrary group. Then
can be given the natural structure of a
-module by putting, for
,
,
, where
is an inverse image of
in
.
When the group operation in the Abelian group is written multiplicatively (for example, if
is the multiplicative group of a field), the notation
is also used instead of
, that is, the action of
is written exponentially.
Let a -module
be given. By associating with an element
the automorphism
of
, a homomorphism of
into the group of invertible elements of the ring
is obtained. Conversely, any homomorphims of
into the group of invertible elements of
gives
the structure of a
-module.
The notions of a module over a ring and a -module are closely connected. Namely, any
-module
can be regarded as a module over the group ring (cf. Group algebra)
if the action of
on
is extended linearly, that is, if one puts
![]() |
where ,
,
. Conversely, given a unitary
-module structure on
,
may be regarded to be a
-module.
When is simultaneously a
-module over a commutative ring
and a
-module, where the action of the elements of
on
commutes with the action of the elements of
, then
may be given the structure of a
-module by linearly extending the action from
to
. For example, if
is a linear vector space over a field
, then the specification of a
-module structure on
is equivalent to giving a representation of
in
.
Using the standard involution in
, any left
-module
can be made into a right
-module by putting
for
,
. Similarly, any right
-module can be made into a left
-module.
Modules over a Lie algebra.
Let be a Lie algebra over a commutative ring
and let
be a
-module. The specification of a
-module structure on
consists of the specification of a
-endomorphism
of the group
for each
, where the axiom
![]() |
holds for ,
. This definition differs from that of an
-module given earlier. Giving a
-module structure on
is equivalent to giving a Lie algebra homomorphism of
into the Lie algebra of the ring
. A module over a Lie algebra
may also be regarded as a module in the usual sense over the universal enveloping algebra of
.
Constructions in the theory of modules.
Starting from a given -module it is possible to obtain new
-modules by a number of standard constructions. Thus, with any module
is associated the lattice of its submodules. For example, if
is considered as left module over itself, then its left submodules are precisely the left ideals in
. A number of important types of modules are defined in terms of the lattice of submodules. For example, the condition for termination of a descending (ascending) chain of submodules defines Artinian modules (respectively, Noetherian modules, cf. Artinian module; Noetherian module). The condition for absence of non-trivial submodules, that is, submodules other than 0 or
, defines irreducible or simple modules (cf. Irreducible module).
For a module and any submodule
, the quotient group
can be given the structure of an
-module. This module is called the quotient module of
over
.
A homomorphism of -modules is defined as an Abelian group homomorphism
commuting with multiplication by elements of
, that is,
for all
,
. If two homomorphisms
are given, then their sum, defined by
, is again a homomorphism of
-modules. This addition gives an Abelian group structure to the set
of all homomorphisms of
into
. For any homomorphism
the submodules
(the kernel of
) and
(the image of
), and also the quotient modules
(the cokernel of
) and
(the coimage of
) are defined. The modules
and
are canonically isomorphic and therefore usually identified. For example, for any left ideal
of
the quotient module
is defined. The module
is irreducible if and only if
is a maximal left ideal (cf. Maximal ideal). If
is an irreducible
-module not annihilating the ring
, then
is isomorphic to
for some maximal left ideal
.
For any family of -modules
, where
runs through some index set
, the direct sum and direct product of
exist in the category of
-modules. Here an element of the direct product
may be interpreted as a vector
the components of which are indexed by
and where for each
,
. The sum of such vectors and their multiplication by elements of the ring are defined componentwise. The direct sum
of the family
can be interpreted as the submodule of the direct product consisting of the vectors all components of which, except for finitely many, are equal to zero.
For a projective (inductive) system of -modules the projective (inductive) limit of this system can be naturally equipped with the structure of an
-module. The direct product and direct sum may be considered as special cases of the notions of a projective and an inductive limit.
Generators and relations.
Let be a subset of an
-module
. The submodule generated by
is the intersection of the submodules of
which contain
. If this submodule coincides with
, then
is called a family (system) of generators of the module
. A module admitting a finite family of generators is called a finitely-generated module. For example, in a Noetherian ring any ideal is a finitely-generated module. A direct sum of a finite number of finitely-generated modules is again finitely generated. Any quotient module of a finitely-generated module is also finitely generated. For the construction of a system of generators for a module
, the Nakayama lemma often turns out to be useful: For any ideal
contained in the radical of a ring
the condition
implies
. In particular, under the conditions of Nakayama's lemma elements
form a system of generators for
if their images generate the module
. This is used particularly often when
is a local ring and
is the maximal ideal in
.
Let be a module with system of generators
. Then a mapping
defines an epimorphism of the free
-module
with generators
onto
(
can be defined as the set of formal finite sums
,
, and
is extended from the generators to
by linearity). The elements of
are called relations between the generators
of
. If
can be represented as a quotient module of a finitely-generated free module
so that the module of relations
is also finitely generated, then
is called a finitely-presented module. For example, over a Noetherian ring
any finitely-generated module is finitely presented. In general, being finitely generated does not imply being finitely presented.
Change of rings.
There are standard constructions which allow an -module
to be considered as a module over some other ring. For example, let
be a homomorphism of rings. Then, putting
,
can be considered as a
-module. The resulting
-module is said to be obtained by base change or, in particular in the case that
is a subring of
, by restriction of scalars. If
is a unitary
-module and
takes the identity to the identity,
becomes a unitary
-module.
Let a ring homomorphism and an
-module
be given. Then
may be given the structure of a
-module (cf. Bimodule) by putting
for
,
, and the left
-module
can be considered. One says that this module is obtained from
by extension of scalars.
The category of modules.
The class of all modules over a given ring with homomorphisms of modules as morphisms forms an Abelian category, denoted, for instance, by
-mod or
. The most important functors defined on this category are
(homomorphism) and
(tensor product). The functor
takes values in the category of Abelian groups and associates to a pair of
-modules
the group
. For
and
the mappings
![]() |
and
![]() |
are defined in the obvious way; that is, the functor is contravariant in its first argument and covariant in the second. When
or
carry a bimodule structure, the group
has an additional module structure. If
is an
-module,
is a right
-module and if
is an
-module, then
is a left
-module.
The functor takes a pair
, where
is a right
-module and
is a left
-module, to the tensor product
of
and
over
. This functor takes values in the category of Abelian groups and is covariant with respect to both
and
. When
or
is a bimodule, the group
may be equipped with an additional structure. Namely, if
is a
-module,
is a
-module, and if
is an
-module, then
is a right
-module. The study of the functors
and
, and also of their derived functors, is one of the fundamental problems of homological algebra.
Many important types of modules can be characterized in terms of and
. Thus, a projective module
is defined by the requirement that the functor
(as a functor in
) is exact (cf. Exact functor). Similarly, an injective module
is defined by the requirement of exactness of
(in
). A flat module
is defined by the requirement of exactness of the functor
.
A module over a given ring can be considered from two points of view.
A) Modules can be studied from the point of view of their intrinsic structure. The fundamental problem here is the complete classification of modules, that is, the construction for each module of a system of invariants which characterizes the module up to an isomorphism, and, given a set of invariants, the ability to construct a module with those invariants. For certain types of rings such a description is possible. For example, if is a finitely-generated module over the group ring
of a finite group
, where
is a field of characteristic coprime with the order of
, then
is representable as a finite direct sum of irreducible submodules (
is completely reducible, cf. Completely-reducible module). This representation is unique up to an isomorphism (the choice of the irreducible modules is, in general, not unique). All irreducible submodules also admit a simple description: All of them are contained in the regular representation of
and are in one-to-one correspondence with the irreducible characters of the group. Modules over principal ideal rings and over Dedekind rings also have a simple description. Namely, any finitely-generated module
over a principal ideal ring
is isomorphic to a finite direct sum of modules of the form
, where
are ideals of
(possibly null), and where
. The ideals
are uniquely determined by this last condition. Thus, the set of invariants
completely determines
. If
is a finitely-generated module over a Dedekind ring
, then
, where
is a torsion module (periodic module) and
is a torsion-free module (the choice of
is not unique). The module
is annihilated by some ideal
of
and, consequently, is a module over the principal ideal ring
and admits the description given above;
is representable in the form
, where
is an ideal of
and
is the
-fold direct sum. The module
is, up to an isomorphism, determined by two invariants: the number
and the class of
in the ideal class group.
B) Another approach to the study of modules consists of studying the category -mod and in considering a given module
as an object of this category. Such a study is the object of homological algebra and algebraic
-theory. On this route many important and deep results have been found.
Often, modules which carry some extra structure are considered. Thus one considers graded modules, filtered modules, topological modules, modules with a sesquilinear form, etc. (cf. Graded module; Topological module; Filtered module).
References
[1] | N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , 1 , Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French) MR0354207 |
[2] | N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French) MR0360549 Zbl 0279.13001 |
[3] | N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French) MR0682756 Zbl 0319.17002 |
[4] | S. Lang, "Algebra" , Addison-Wesley (1984) MR0783636 Zbl 0712.00001 |
[5] | B.L. van der Waerden, "Algebra" , 1–2 , Springer (1967–1971) (Translated from German) MR1541390 Zbl 1032.00002 Zbl 1032.00001 Zbl 0903.01009 Zbl 0781.12003 Zbl 0781.12002 Zbl 0724.12002 Zbl 0724.12001 Zbl 0569.01001 Zbl 0534.01001 Zbl 0997.00502 Zbl 0997.00501 Zbl 0316.22001 Zbl 0297.01014 Zbl 0221.12001 Zbl 0192.33002 Zbl 0137.25403 Zbl 0136.24505 Zbl 0087.25903 Zbl 0192.33001 Zbl 0067.00502 |
[6] | A.I. Kostrikin, "Introduction to algebra" , Springer (1982) (Translated from Russian) MR0661256 Zbl 0482.00001 |
[7] | N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956) MR0081264 Zbl 0073.02002 |
[8] | I.N. Herstein, "Noncommutative rings" , Math. Assoc. Amer. (1968) MR1535024 MR0227205 Zbl 0177.05801 |
[9] | C. Faith, "Algebra: rings, modules and categories" , 1–2 , Springer (1981–1976) MR0551052 MR0491784 MR0366960 Zbl 0508.16001 Zbl 0266.16001 |
[10] | H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956) MR0077480 Zbl 0075.24305 |
[11] | S. MacLane, "Homology" , Springer (1963) Zbl 0818.18001 Zbl 0328.18009 |
[12] | H. Bass, "Algebraic ![]() |
[13] | J.W. Milnor, "Introduction to algebraic ![]() |
Module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Module&oldid=42236