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 module is a generalization of the notion of
vector space in which the
field of
scalars
Scalar may refer to:
* Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
* Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
is replaced by a
ring. The concept of ''module'' generalizes also the notion of
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
, since the abelian groups are exactly the modules over the ring of
integers.
Like a vector space, a module is an additive abelian group, and scalar multiplication is
distributive over the operation of addition between elements of the ring or module and is
compatible with the ring multiplication.
Modules are very closely related to the
representation theory of
groups. They are also one of the central notions of
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...
and
homological algebra, and are used widely in
algebraic geometry and
algebraic topology.
Introduction and definition
Motivation
In a vector space, the set of
scalars
Scalar may refer to:
* Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
* Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
is a
field and acts on the vectors by scalar multiplication, subject to certain axioms such as the
distributive law. In a module, the scalars need only be a
ring, so the module concept represents a significant generalization. In commutative algebra, both
ideals and
quotient rings are modules, so that many arguments about ideals or quotient rings can be combined into a single argument about modules. In non-commutative algebra, the distinction between left ideals, ideals, and modules becomes more pronounced, though some ring-theoretic conditions can be expressed either about left ideals or left modules.
Much of the theory of modules consists of extending as many of the desirable properties of vector spaces as possible to the realm of modules over a "
well-behaved" ring, such as a
principal ideal domain. However, modules can be quite a bit more complicated than vector spaces; for instance, not all modules have a
basis, and even those that do,
free modules, need not have a unique
rank if the underlying ring does not satisfy the
invariant basis number condition, unlike vector spaces, which always have a (possibly infinite) basis whose cardinality is then unique. (These last two assertions require the
axiom of choice in general, but not in the case of finite-dimensional spaces, or certain well-behaved infinite-dimensional spaces such as
L''p'' spaces.)
Formal definition
Suppose that ''R'' is a
ring, and 1 is its multiplicative identity.
A left ''R''-module ''M'' consists of an
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
and an operation such that for all ''r'', ''s'' in ''R'' and ''x'', ''y'' in ''M'', we have
#
#
#
#
The operation · is called ''scalar multiplication''. Often the symbol · is omitted, but in this article we use it and reserve juxtaposition for multiplication in ''R''. One may write
''R''''M'' to emphasize that ''M'' is a left ''R''-module. A right ''R''-module ''M''
''R'' is defined similarly in terms of an operation .
Authors who do not require rings to be
unital omit condition 4 in the definition above; they would call the structures defined above "unital left ''R''-modules". In this article, consistent with the
glossary of ring theory, all rings and modules are assumed to be unital.
An (''R'',''S'')-
bimodule is an abelian group together with both a left scalar multiplication · by elements of ''R'' and a right scalar multiplication ∗ by elements of ''S'', making it simultaneously a left ''R''-module and a right ''S''-module, satisfying the additional condition for all ''r'' in ''R'', ''x'' in ''M'', and ''s'' in ''S''.
If ''R'' is
commutative, then left ''R''-modules are the same as right ''R''-modules and are simply called ''R''-modules.
Examples
*If ''K'' is a
field, then ''K''-
vector spaces (vector spaces over ''K'') and ''K''-modules are identical.
*If ''K'' is a field, and ''K''
'x''a univariate
polynomial ring, then a
''K'' module">'x''module ''M'' is a ''K''-module with an additional action of ''x'' on ''M'' that commutes with the action of ''K'' on ''M''. In other words, a ''K''
'x''module is a ''K''-vector space ''M'' combined with a
linear map from ''M'' to ''M''. Applying the
to this example shows the existence of the
rational and
Jordan canonical forms.
*The concept of a Z-module agrees with the notion of an abelian group. That is, every
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
is a module over the ring of
integers Z in a unique way. For , let (''n'' summands), , and . Such a module need not have a
basis—groups containing
torsion elements do not. (For example, in the group of integers
modulo
In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation).
Given two positive numbers and , modulo (often abbreviated as ) is ...
3, one cannot find even one element which satisfies the definition of a linearly independent set since when an integer such as 3 or 6 multiplies an element, the result is 0. However, if a
finite field is considered as a module over the same finite field taken as a ring, it is a vector space and does have a basis.)
*The
decimal fractions (including negative ones) form a module over the integers. Only
singletons are linearly independent sets, but there is no singleton that can serve as a basis, so the module has no basis and no rank.
*If ''R'' is any ring and ''n'' a
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
, then the
cartesian product ''R''
''n'' is both a left and right ''R''-module over ''R'' if we use the component-wise operations. Hence when , ''R'' is an ''R''-module, where the scalar multiplication is just ring multiplication. The case yields the trivial ''R''-module consisting only of its identity element. Modules of this type are called
free and if ''R'' has
invariant basis number (e.g. any commutative ring or field) the number ''n'' is then the rank of the free module.
*If M
''n''(''R'') is the ring of
matrices over a ring ''R'', ''M'' is an M
''n''(''R'')-module, and ''e''
''i'' is the matrix with 1 in the -entry (and zeros elsewhere), then ''e''
''i''''M'' is an ''R''-module, since . So ''M'' breaks up as the direct sum of ''R''-modules, . Conversely, given an ''R''-module ''M''
0, then ''M''
0⊕''n'' is an M
''n''(''R'')-module. In fact, the
category of ''R''-modules and the
category of M
''n''(''R'')-modules are
equivalent. The special case is that the module ''M'' is just ''R'' as a module over itself, then ''R''
''n'' is an M
''n''(''R'')-module.
*If ''S'' is a
nonempty 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 ...
, ''M'' is a left ''R''-module, and ''M''
''S'' is the collection of all
functions , then with addition and scalar multiplication in ''M''
''S'' defined pointwise by and , ''M''
''S'' is a left ''R''-module. The right ''R''-module case is analogous. In particular, if ''R'' is commutative then the collection of ''R-module homomorphisms'' (see below) is an ''R''-module (and in fact a ''submodule'' of ''N''
''M'').
*If ''X'' is a
smooth manifold, then the
smooth functions from ''X'' to the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s form a ring ''C''
∞(''X''). The set of all smooth
vector fields defined on ''X'' form a module over ''C''
∞(''X''), and so do the
tensor fields and the
differential forms on ''X''. More generally, the sections of any
vector bundle form a
projective module over ''C''
∞(''X''), and by
Swan's theorem, every projective module is isomorphic to the module of sections of some bundle; the
category of ''C''
∞(''X'')-modules and the category of vector bundles over ''X'' are
equivalent.
*If ''R'' is any ring and ''I'' is any
left ideal in ''R'', then ''I'' is a left ''R''-module, and analogously right ideals in ''R'' are right ''R''-modules.
*If ''R'' is a ring, we can define the
opposite ring ''R''
op which has the same
underlying set
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set ...
and the same addition operation, but the opposite multiplication: if in ''R'', then in ''R''
op. Any ''left'' ''R''-module ''M'' can then be seen to be a ''right'' module over ''R''
op, and any right module over ''R'' can be considered a left module over ''R''
op.
*
Modules over a Lie algebra are (associative algebra) modules over its
universal enveloping algebra.
*If ''R'' and ''S'' are rings with a
ring homomorphism , then every ''S''-module ''M'' is an ''R''-module by defining . In particular, ''S'' itself is such an ''R''-module.
Submodules and homomorphisms
Suppose ''M'' is a left ''R''-module and ''N'' is a
subgroup of ''M''. Then ''N'' is a submodule (or more explicitly an ''R''-submodule) if for any ''n'' in ''N'' and any ''r'' in ''R'', the product (or for a right ''R''-module) is in ''N''.
If ''X'' is any
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
of an ''R''-module, then the submodule spanned by ''X'' is defined to be
where ''N'' runs over the submodules of ''M'' which contain ''X'', or explicitly
, which is important in the definition of tensor products.
The set of submodules of a given module ''M'', together with the two binary operations + and ∩, forms a
lattice which satisfies the
modular law
In the branch of mathematics called order theory, a modular lattice is a lattice that satisfies the following self-dual condition,
;Modular law: implies
where are arbitrary elements in the lattice, ≤ is the partial order, and &n ...
:
Given submodules ''U'', ''N''
1, ''N''
2 of ''M'' such that , then the following two submodules are equal: .
If ''M'' and ''N'' are left ''R''-modules, then a
map is a
homomorphism of ''R''-modules if for any ''m'', ''n'' in ''M'' and ''r'', ''s'' in ''R'',
:
.
This, like any
homomorphism
In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
of mathematical objects, is just a mapping which preserves the structure of the objects. Another name for a homomorphism of ''R''-modules is an ''R''-
linear map.
A
bijective module homomorphism is called a module
isomorphism, and the two modules ''M'' and ''N'' are called isomorphic. Two isomorphic modules are identical for all practical purposes, differing solely in the notation for their elements.
The
kernel of a module homomorphism is the submodule of ''M'' consisting of all elements that are sent to zero by ''f'', and the
image of ''f'' is the submodule of ''N'' consisting of values ''f''(''m'') for all elements ''m'' of ''M''. The
isomorphism theorem
In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exis ...
s familiar from groups and vector spaces are also valid for ''R''-modules.
Given a ring ''R'', the set of all left ''R''-modules together with their module homomorphisms forms an
abelian category, denoted by ''R''-Mod (see
category of modules).
Types of modules
; Finitely generated: An ''R''-module ''M'' is
finitely generated if there exist finitely many elements ''x''
1, ..., ''x''
''n'' in ''M'' such that every element of ''M'' is a
linear combination of those elements with coefficients from the ring ''R''.
; Cyclic: A module is called a
cyclic module if it is generated by one element.
; Free: A
free ''R''-module is a module that has a basis, or equivalently, one that is isomorphic to a
direct sum of copies of the ring ''R''. These are the modules that behave very much like vector spaces.
; Projective:
Projective modules are
direct summands of free modules and share many of their desirable properties.
; Injective:
Injective modules are defined dually to projective modules.
; Flat: A module is called
flat if taking the
tensor product of it with any
exact sequence of ''R''-modules preserves exactness.
; Torsionless: A module is called
torsionless if it embeds into its algebraic dual.
; Simple: A
simple module ''S'' is a module that is not and whose only submodules are and ''S''. Simple modules are sometimes called ''irreducible''.
[Jacobson (1964)]
p. 4
Def. 1;
; Semisimple: A
semisimple module
In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts. A ring that is a semisimple module over itself ...
is a direct sum (finite or not) of simple modules. Historically these modules are also called ''completely reducible''.
; Indecomposable: An
indecomposable module is a non-zero module that cannot be written as a
direct sum of two non-zero submodules. Every simple module is indecomposable, but there are indecomposable modules which are not simple (e.g.
uniform module In abstract algebra, a module is called a uniform module if the intersection of any two nonzero submodules is nonzero. This is equivalent to saying that every nonzero submodule of ''M'' is an essential submodule. A ring may be called a right (lef ...
s).
; Faithful: A
faithful module ''M'' is one where the action of each in ''R'' on ''M'' is nontrivial (i.e. for some ''x'' in ''M''). Equivalently, the
annihilator of ''M'' is the
zero ideal.
; Torsion-free: A
torsion-free module
In algebra, a torsion-free module is a module over a ring such that zero is the only element annihilated by a regular element (non zero-divisor) of the ring. In other words, a module is ''torsion free'' if its torsion submodule is reduced to its ...
is a module over a ring such that 0 is the only element annihilated by a regular element (non
zero-divisor
In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zero ...
) of the ring, equivalently implies or .
; Noetherian: A
Noetherian module is a module which satisfies the
ascending chain condition In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly ideals in certain commutative rings.Jacobson (2009), p. 142 and 147 These c ...
on submodules, that is, every increasing chain of submodules becomes stationary after finitely many steps. Equivalently, every submodule is finitely generated.
; Artinian: An
Artinian module is a module which satisfies the
descending chain condition on submodules, that is, every decreasing chain of submodules becomes stationary after finitely many steps.
; Graded: A
graded module is a module with a decomposition as a direct sum over a
graded ring such that for all ''x'' and ''y''.
; Uniform: A
uniform module In abstract algebra, a module is called a uniform module if the intersection of any two nonzero submodules is nonzero. This is equivalent to saying that every nonzero submodule of ''M'' is an essential submodule. A ring may be called a right (lef ...
is a module in which all pairs of nonzero submodules have nonzero intersection.
Further notions
Relation to representation theory
A representation of a group ''G'' over a field ''k'' is a module over the
group ring ''k''
'G''
If ''M'' is a left ''R''-module, then the ''action'' of an element ''r'' in ''R'' is defined to be the map that sends each ''x'' to ''rx'' (or ''xr'' in the case of a right module), and is necessarily a
group endomorphism of the abelian group . The set of all group endomorphisms of ''M'' is denoted End
Z(''M'') and forms a ring under addition and
composition, and sending a ring element ''r'' of ''R'' to its action actually defines a
ring homomorphism from ''R'' to End
Z(''M'').
Such a ring homomorphism is called a ''representation'' of ''R'' over the abelian group ''M''; an alternative and equivalent way of defining left ''R''-modules is to say that a left ''R''-module is an abelian group ''M'' together with a representation of ''R'' over it. Such a representation may also be called a ''ring action'' of ''R'' on ''M''.
A representation is called ''faithful'' if and only if the map is
injective. In terms of modules, this means that if ''r'' is an element of ''R'' such that for all ''x'' in ''M'', then . Every abelian group is a faithful module over the
integers or over some
ring of integers modulo ''n'', Z/''n''Z.
Generalizations
A ring ''R'' corresponds to a
preadditive category R with a single
object
Object may refer to:
General meanings
* Object (philosophy), a thing, being, or concept
** Object (abstract), an object which does not exist at any particular time or place
** Physical object, an identifiable collection of matter
* Goal, an ...
. With this understanding, a left ''R''-module is just a covariant
additive functor from R to the
category Ab of abelian groups, and right ''R''-modules are contravariant additive functors. This suggests that, if C is any preadditive category, a covariant additive functor from C to Ab should be considered a generalized left module over C. These functors form a
functor category C-Mod which is the natural generalization of the module category ''R''-Mod.
Modules over ''commutative'' rings can be generalized in a different direction: take a
ringed space
In mathematics, a ringed space is a family of ( commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf ...
(''X'', O
''X'') and consider the
sheaves of O
''X''-modules (see
sheaf of modules In mathematics, a sheaf of ''O''-modules or simply an ''O''-module over a ringed space (''X'', ''O'') is a sheaf ''F'' such that, for any open subset ''U'' of ''X'', ''F''(''U'') is an ''O''(''U'')-module and the restriction maps ''F''(''U'') ...
). These form a category O
''X''-Mod, and play an important role in modern
algebraic geometry. If ''X'' has only a single point, then this is a module category in the old sense over the commutative ring O
''X''(''X'').
One can also consider modules over a
semiring. Modules over rings are abelian groups, but modules over semirings are only
commutative monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoid ...
s. Most applications of modules are still possible. In particular, for any
semiring ''S'', the matrices over ''S'' form a semiring over which the tuples of elements from ''S'' are a module (in this generalized sense only). This allows a further generalization of the concept of
vector space incorporating the semirings from theoretical computer science.
Over
near-rings, one can consider near-ring modules, a nonabelian generalization of modules.
See also
*
Group ring
*
Algebra (ring theory)
*
Module (model theory)
*
Module spectrum
*
Annihilator
Notes
References
* F.W. Anderson and K.R. Fuller: ''Rings and Categories of Modules'', Graduate Texts in Mathematics, Vol. 13, 2nd Ed., Springer-Verlag, New York, 1992, ,
*
Nathan Jacobson. ''Structure of rings''. Colloquium publications, Vol. 37, 2nd Ed., AMS Bookstore, 1964,
External links
*
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