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An Abelian group with the distributive action of a ring. A module is a generalization of a (linear) vector space over a field $K$, when $K$ is replaced by a ring.

Let a ring $A$ be given. An additive Abelian group $M$ is called a left $A$-module if there is a mapping $A\times M \to M$ whose value on a pair $(a, m)$, for $a \in A$, $m \in M$, written $am$, satisfies the axioms:

1) $a(m_1 + m_2) = am_1 + am_2$;

2) $(a_1 + a_2)m = a_1 m + a_2 m$;

3) $a_1(a_2 m) = (a_1 a_2) m$. If $A$ is a ring with identity, then it is usual to require in addition that for any $m \in M$, $1m = m$. A module with this property is called unitary or unital (cf. Unitary module).

Right $A$-modules are defined similarly; axiom 3) is replaced by $(ma_1)a_2 = m(a_1 a_2)$. Any right $A$-module can be considered as a left $A^\text{opp}$-module over the opposite ring $A^\text{opp}$ anti-isomorphic to $A$; hence, corresponding to any result about right $A$-modules there is a result about left $A^\text{opp}$-modules, and conversely. When $A$ is commutative, any left $A$-module can be considered as a right $A$-module and the distinction between left and right modules disappears. Below only left $A$-modules are discussed.

The simplest examples of modules (finite Abelian groups; they are $\ZZ$-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 $M$ is a module over the ring of integers $\ZZ$. For $a \in \ZZ$ and $m \in M$ the product $am$ is defined as the result of adding $m$ to itself $a$ times.

2) When $A$ is a field, the notion of a unitary $A$-module is exactly equivalent to the notion of a linear vector space over $A$.

3) An $n$-dimensional vector space $V$ over a field $K$ (provided with coordinates) can be considered as a module over the ring $M_n(K)$ of all $(n\times n)$-matrices with coefficients in $K$. For $v \in V$ and $X \in M_n(K)$ the product $Xv$ is defined as multiplication of the matrix $X$ by the column of coordinates of the vector $v$.

4) An associative ring (cf. Associative rings and algebras) $A$ is a left $A$-module. Multiplication of elements of the ring by elements of the module is ordinary multiplication in $A$.

5) The set of differential forms on a smooth manifold $X$ has the natural structure of a module over the ring of all smooth functions on $X$.

6) Connected with any Abelian group $M$ is the associative ring with identity, $\End(M)$, of all endomorphisms of $M$. The group $M$ has a natural $\End(M)$-module structure.

If there is an $A$-module structure on $M$, for some ring $A$, then the mapping $m \mapsto am$ is an endomorphism of $M$ for any $a \in A$. Associating with the element $a \in A$ the endomorphism of $M$ that it generates, one obtains a homomorphism $\phi$ of $A$ into $\End(M)$. Conversely, any homomorphism $\phi: A \to \End(M)$ defines the structure of an $A$-module on $M$. Thus, the specification of an $A$-module structure on an Abelian group $M$ is equivalent to the specification of a homomorphism of rings $\phi: A \to \End(M)$. Such a homomorphism is also called a representation of the ring $A$, and $M$ is called a representation module. Connected with any representation $\phi$ is a two-sided ideal $\Ann(M) = \Ker \phi$, consisting of the $a \in A$ such that $am = 0$ for all $m \in M$. This ideal is called the annihilator of the module $M$. When $\Ann(M) = 0$, the representation is called faithful and $M$ is called a faithful module (or faithful representation).

It is obvious that a module $M$ can also be considered as a module over the quotient ring $A/\Ann(M)$. In particular, although the definition of a module does not assume the associativity of $A$, the ring $A/\Ann(M)$ is always associative. Therefore, in the majority of cases the discussion may be restricted to modules over associative rings. Everywhere below, unless stated otherwise, $A$ is assumed to be associative.

$G$-modules.

Let $G$ be a group. An additive Abelian group $M$ is called a left $G$-module if there is a mapping $G\times M \to M$ whose value at a pair $(g, m)$, where $g \in G$, $m \in M$, is written as $gm$, and where for any $g \in G$ the mapping $m \mapsto gm$ is an endomorphism of $M$; for any $g_1, g_2 \in G$, $m \in M$, $(g_1 g_2)m = g_1(g_2 m)$; and for all $m \in M$, $1m = m$, where 1 is the identity of $G$. For any $g \in G$ the mapping $m \mapsto gm$ is an automorphism of the group $M$.

Right $G$-modules may be defined similarly.

Examples of $K$-modules.

1) Let $k$ be a Galois extension of a field $G$ with Galois group $K$. Then the additive and multiplicative groups of $K$ have the natural structure of $G$-modules. If $k$ is an algebraic number field, then other $G$-modules are: the additive group of the ring of integers of $K$, the group of units of $K$, the group of divisors and the divisor class group of $K$, etc. A module over a Galois group is called a Galois module.

2) Let an extension of an Abelian group $M$ be given, that is, an exact sequence of groups

$$1 \too M \too F \too G \too 1,$$ where $M$ is an Abelian normal subgroup of $F$ and $G$ is an arbitrary group. Then $M$ can be given the natural structure of a $G$-module by putting, for $g \in G$, $m \in M$, $gm = \overline g m \overline g\inv$, where $\overline g$ is an inverse image of $g$ in $F$.

When the group operation in the Abelian group $M$ is written multiplicatively (for example, if $M$ is the multiplicative group of a field), the notation $m^g$ is also used instead of $gm$, that is, the action of $G$ is written exponentially.

Let a $G$-module $M$ be given. By associating with an element $g \in G$ the automorphism $m \mapsto gm$ of $M$, a homomorphism of $G$ into the group of invertible elements of the ring $\End(M)$ is obtained. Conversely, any homomorphims of $G$ into the group of invertible elements of $\End(M)$ gives $M$ the structure of a $G$-module.

The notions of a module over a ring and a $G$-module are closely connected. Namely, any $G$-module $M$ can be regarded as a module over the group ring (cf. Group algebra) $\ZZ G$ if the action of $G$ on $M$ is extended linearly, that is, if one puts

$$\left(\sum a_i g_i\right) m = \sum a_i (g_i m),$$ where $a_i \in \ZZ$, $g_i \in G$, $m \in M$. Conversely, given a unitary $\ZZ G$-module structure on $M$, $M$ may be regarded to be a $G$-module.

When $M$ is simultaneously a $K$-module over a commutative ring $K$ and a $G$-module, where the action of the elements of $G$ on $M$ commutes with the action of the elements of $K$, then $M$ may be given the structure of a $KG$-module by linearly extending the action from $G$ to $KG$. For example, if $V$ is a linear vector space over a field $K$, then the specification of a $KG$-module structure on $V$ is equivalent to giving a representation of $G$ in $V$.

Using the standard involution $g \mapsto g\inv$ in $G$, any left $G$-module $M$ can be made into a right $G$-module by putting $mg = g\inv m$ for $m \in M$, $g \in G$. Similarly, any right $G$-module can be made into a left $G$-module.

Modules over a Lie algebra.

Let $\lieg$ be a Lie algebra over a commutative ring $K$ and let $M$ be a $K$-module. The specification of a $\lieg$-module structure on $M$ consists of the specification of a $K$-endomorphism $m \mapsto gm$ of the group $M$ for each $g \in \lieg$, where the axiom

$$[g_1, g_2] m = g_1(g_2 m) - g_2(g_1 m)$$ holds for $g_1, g_2 \in \lieg$, $m \in M$. This definition differs from that of an $A$-module given earlier. Giving a $\lieg$-module structure on $M$ is equivalent to giving a Lie algebra homomorphism of $\lieg$ into the Lie algebra of the ring $\End(M)$. A module over a Lie algebra $\lieg$ may also be regarded as a module in the usual sense over the universal enveloping algebra of $\lieg$.

Constructions in the theory of modules.

Starting from a given $A$-module it is possible to obtain new $A$-modules by a number of standard constructions. Thus, with any module $M$ is associated the lattice of its submodules. For example, if $A$ is considered as left module over itself, then its left submodules are precisely the left ideals in $A$. 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 $M$, defines irreducible or simple modules (cf. Irreducible module).

For a module $M$ and any submodule $N$, the quotient group $M/N$ can be given the structure of an $A$-module. This module is called the quotient module of $M$ over $N$.

A homomorphism of $A$-modules is defined as an Abelian group homomorphism $f : M \to N$ commuting with multiplication by elements of $A$, that is, $f(am) = af(m)$ for all $m \in M$, $a \in A$. If two homomorphisms $f_1, f_2 : M \to N$ are given, then their sum, defined by $(f_1 + f_2)(m) = f_1(m) = f_2(m)$, is again a homomorphism of $A$-modules. This addition gives an Abelian group structure to the set $\Hom_A(M, N)$ of all homomorphisms of $M$ into $N$. For any homomorphism $f : M \to N$ the submodules $\Ker f$ (the kernel of $f$) and $\Im f$ (the image of $f$), and also the quotient modules $\Coker f = N / \Im f$ (the cokernel of $f$) and $\Coim f = M/\Ker f$ (the coimage of $f$) are defined. The modules $\Im f$ and $\Coim f$ are canonically isomorphic and therefore usually identified. For example, for any left ideal $J$ of $A$ the quotient module $A/J$ is defined. The module $A/J$ is irreducible if and only if $J$ is a maximal left ideal (cf. Maximal ideal). If $M$ is an irreducible $A$-module not annihilating the ring $A$, then $M$ is isomorphic to $A/J$ for some maximal left ideal $J$.

For any family of $A$-modules $\{M_i\}$, where $i$ runs through some index set $J$, the direct sum and direct product of $\{M_i\}$ exist in the category of $A$-modules. Here an element of the direct product $\prod_{i \in J} M_i$ may be interpreted as a vector $(ldots, m_i, \ldots)$ the components of which are indexed by $J$ and where for each $i$, $m_i \in M_i$. The sum of such vectors and their multiplication by elements of the ring are defined componentwise. The direct sum $\sum_{i \in J} M_i$ of the family $\{M_i\}$ 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 $A$-modules the projective (inductive) limit of this system can be naturally equipped with the structure of an $A$-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 $X$ be a subset of an $A$-module $M$. The submodule generated by $X$ is the intersection of the submodules of $M$ which contain $X$. If this submodule coincides with $M$, then $X$ is called a family (system) of generators of the module $M$. 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 $M$, the Nakayama lemma often turns out to be useful: For any ideal $\frakA$ contained in the radical of a ring $A$ the condition $\frakA M = M$ implies $M = 0$. In particular, under the conditions of Nakayama's lemma elements $m_1, \ldots, m_r$ form a system of generators for $M$ if their images generate the module $M/\frakA M$. This is used particularly often when $A$ is a local ring and $\frakA$ is the maximal ideal in $A$.

Let $M$ be a module with system of generators $\{x_i\}_{i \in J}$. Then a mapping $\phi : y_i \to x_i$ defines an epimorphism of the free $A$-module $F$ with generators $\{y_i\}_{i \in I}$ onto $M$ ($F$ can be defined as the set of formal finite sums $\sum a_i y_i$, $a_i \in A$, and $\phi$ is extended from the generators to $F$ by linearity). The elements of $R = \Ker \phi$ are called relations between the generators $\{x_i\}$ of $M$. If $M$ can be represented as a quotient module of a finitely-generated free module $F$ so that the module of relations $R$ is also finitely generated, then $M$ is called a finitely-presented module. For example, over a Noetherian ring $A$ 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 $A$-module $M$ to be considered as a module over some other ring. For example, let $\phi : B \to A$ be a homomorphism of rings. Then, putting $bm = \phi(b) m$, $M$ can be considered as a $B$-module. The resulting $B$-module is said to be obtained by base change or, in particular in the case that $B$ is a subring of $A$, by restriction of scalars. If $M$ is a unitary $A$-module and $\phi$ takes the identity to the identity, $M$ becomes a unitary $B$-module.

Let a ring homomorphism $\phi: A \to B$ and an $A$-module $M$ be given. Then $B$ may be given the structure of a $(B, A)$-module (cf. Bimodule) by putting $ba = b\phi(a)$ for $b\in B$, $a \in A$, and the left $B$-module $B \tensor_A M$ can be considered. One says that this module is obtained from $M$ by extension of scalars.

The category of modules.

The class of all modules over a given ring $A$ with homomorphisms of modules as morphisms forms an Abelian category, denoted, for instance, by $A$-mod or $\Mod_A$. The most important functors defined on this category are $\Hom$ (homomorphism) and $\tensor$ (tensor product). The functor $\Hom$ takes values in the category of Abelian groups and associates to a pair of $A$-modules $M, N$ the group $\Hom_A(M, N)$. For $f : M_1 \to M$ and $\phi : N \to N_1$ the mappings

$$f' : \Hom_A(M, N) \to \Hom_A(M_1, N)$$ and

$$\phi' : \Hom_A(M, N) \to \Hom_A(M, N_1)$$ are defined in the obvious way; that is, the functor $\Hom$ is contravariant in its first argument and covariant in the second. When $M$ or $N$ carry a bimodule structure, the group $\Hom_A(M,N)$ has an additional module structure. If $N$ is an $(A, B)$-module, $\Hom_A(M,N)$ is a right $B$-module and if $M$ is an $(A,B)$-module, then $\Hom_A(M, N)$ is a left $B$-module.

The functor $\tensor_A$ takes a pair $M, N$, where $M$ is a right $A$-module and $N$ is a left $A$-module, to the tensor product $M\tensor_A N$ of $M$ and $N$ over $A$. This functor takes values in the category of Abelian groups and is covariant with respect to both $M$ and $N$. When $M$ or $N$ is a bimodule, the group $M \tensor_A N$ may be equipped with an additional structure. Namely, if $M$ is a $(B, A)$-module, $M\tensor_A N$ is a $B$-module, and if $N$ is an $(A, B)$-module, then $M\tensor_A N$ is a right $B$-module. The study of the functors $\Hom$ and $\tensor$, 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 $\Hom$ and $\tensor$. Thus, a projective module $M$ is defined by the requirement that the functor $\Hom_A(M, X)$ (as a functor in $X$) is exact (cf. Exact functor). Similarly, an injective module $N$ is defined by the requirement of exactness of $\Hom_A(X, N)$ (in $X$). A flat module $M$ is defined by the requirement of exactness of the functor $M \tensor_A X$.

A module over a given ring $A$ 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 $M$ is a finitely-generated module over the group ring $KG$ of a finite group $G$, where $K$ is a field of characteristic coprime with the order of $G$, then $M$ is representable as a finite direct sum of irreducible submodules ($M$ 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 $G$ 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 $M$ over a principal ideal ring $A$ is isomorphic to a finite direct sum of modules of the form $A/\frakA_i$, where $\frakA_i$ are ideals of $A$ (possibly null), and where $\frakA_1 \subseteq \cdots \subseteq \frakA_m \ne A$. The ideals $\frakA_i$ are uniquely determined by this last condition. Thus, the set of invariants $\{\frakA_i\}$ completely determines $M$. If $M$ is a finitely-generated module over a Dedekind ring $A$, then $M = M_1 \oplus M_2$, where $M_2$ is a torsion module (periodic module) and $M_1$ is a torsion-free module (the choice of $M_1$ is not unique). The module $M_2$ is annihilated by some ideal $\frakA$ of $A$ and, consequently, is a module over the principal ideal ring $A/\frakA$ and admits the description given above; $M_1$ is representable in the form $(\bigoplus^n A) \oplus \frakB$, where $\frakB$ is an ideal of $A$ and $\bigoplus^n$ is the $n$-fold direct sum. The module $M_1$ is, up to an isomorphism, determined by two invariants: the number $n$ and the class of $\frakB$ in the ideal class group.

B) Another approach to the study of modules consists of studying the category $A$-mod and in considering a given module $M$ as an object of this category. Such a study is the object of homological algebra and algebraic $K$-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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