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:
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.
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
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).
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