Module (mathematics)

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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...
, a module is a generalization of the notion of
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
, wherein the
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 grassl ...
of scalars is replaced by a ring. The concept of ''module'' is also a generalization of the one of
abelian group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
, since the abelian groups are exactly the modules over the ring of
integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of t ...
s. 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 Compatibility may refer to: Computing * Backward compatibility, in which newer devices can understand data generated by older devices * Compatibility card, an expansion card for hardware emulation of another device * Compatibility layer, compone ...
with the ring multiplication. Modules are very closely related to the
representation theory Representation theory is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change ( ...
of
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
s. They are also one of the central notions of
commutative algebra Commutative algebra is the branch of algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry ...
and
homological algebra Homological algebra is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (ma ...
, and are used widely in
algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

and
algebraic topology 250px, A torus, one of the most frequently studied objects in algebraic topology Algebraic topology is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathemat ...
.

# Introduction and definition

## Motivation

In a vector space, the set of scalars is 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 grassl ...
and acts on the vectors by scalar multiplication, subject to certain axioms such as the
distributive law In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. In a module, the scalars need only be a ring, so the module concept represents a significant generalization. In commutative algebra, both
ideals Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considered ...
and
quotient ring In ring theory In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studie ...
s 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 In mathematics, a pathological object is one which possesses deviant, irregular or counterintuitive property, in such a way that distinguishes it from what is conceived as a typical object in the same category. The opposite of pathological is ...
" ring, such as a
principal ideal domain In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
. However, modules can be quite a bit more complicated than vector spaces; for instance, not all modules have a basis (linear algebra), basis, and even those that do, free modules, need not have a unique Free_module#Definition, 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 Lp space, 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
and an operation such that for all ''r'', ''s'' in ''R'' and ''x'', ''y'' in ''M'', we have #$r \cdot \left( x + y \right) = r \cdot x + r \cdot y$ #$\left( r + s \right) \cdot x = r \cdot x + s \cdot x$ #$\left( r s \right) \cdot x = r \cdot \left( s \cdot x \right)$ #$1 \cdot x = x .$ 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 algebra, 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 $\left(r \cdot x\right) \ast s = r \cdot \left( x \ast s \right)$ for all ''r'' in ''R'', ''x'' in ''M'', and ''s'' in ''S''. If ''R'' is commutative ring, commutative, then left ''R''-modules are the same as right ''R''-modules and are simply called ''R''-modules.

# Examples

*If ''K'' is 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 grassl ...
, then ''K''-
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s (vector spaces over ''K'') and ''K''-modules are identical. *If ''K'' is a field, and ''K''[''x''] a univariate polynomial ring, then a Polynomial ring#Modules, ''K''[''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 structure theorem for finitely generated modules over a principal ideal domain to this example shows the existence of the Rational canonical form, rational and Jordan normal form, Jordan canonical forms. *The concept of a Z-module agrees with the notion of an abelian group. That is, every
abelian group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
is a module over the ring of
integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of t ...
s Z in a unique way. For , let (''n'' summands), , and . Such a module need not have a basis (linear algebra), basis—groups containing torsion elements do not. (For example, in the group of integers modular arithmetic, modulo 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 singleton (mathematics), 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, 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 module, 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 matrix (mathematics), 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 modules, category of ''R''-modules and the category (mathematics), category of M''n''(''R'')-modules are equivalence of categories, 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 empty set, nonempty Set (mathematics), set, ''M'' is a left ''R''-module, and ''M''''S'' is the collection of all function (mathematics), 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 numbers 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 (mathematics), category of ''C''(''X'')-modules and the category of vector bundles over ''X'' are equivalence of categories, equivalent. *If ''R'' is any ring and ''I'' is any ring ideal, 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 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. * Glossary of Lie algebras#Representation theory, Modules over a Lie algebra are (associative algebra) modules over its universal enveloping algebra. *If ''R'' and ''S'' are rings with a ring homomorphism ''φ'' : ''R'' → ''S'', then every ''S''-module ''M'' is an ''R''-module by defining ''rm'' = ''φ''(''r'')''m''. 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 of an ''R''-module, then the submodule spanned by ''X'' is defined to be $\langle X \rangle = \,\bigcap_ N$ where ''N'' runs over the submodules of ''M'' which contain ''X'', or explicitly $\left\$, which is important in the definition of tensor products. The set of submodules of a given module ''M'', together with the two binary operations + and ∩, forms a Lattice (order), lattice which satisfies the modular lattice, modular law: Given submodules ''U'', ''N''1, ''N''2 of ''M'' such that , then the following two submodules are equal: . If ''M'' and ''N'' are left ''R''-modules, then a map (mathematics), map is a module homomorphism, homomorphism of ''R''-modules if for any ''m'', ''n'' in ''M'' and ''r'', ''s'' in ''R'', :$f\left(r \cdot m + s \cdot n\right) = r \cdot f\left(m\right) + s \cdot f\left(n\right)$. This, like any homomorphism 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 (algebra), kernel of a module homomorphism is the submodule of ''M'' consisting of all elements that are sent to zero by ''f'', and the image (mathematics), image of ''f'' is the submodule of ''N'' consisting of values ''f''(''m'') for all elements ''m'' of ''M''. The isomorphism theorems 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 module, 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 module, free ''R''-module is a module that has a basis, or equivalently, one that is isomorphic to a direct sum of modules, 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 module, flat if taking the tensor product of modules, tensor product of it with any exact sequence of ''R''-modules preserves exactness. ; Torsionless: A module is called torsionless module, 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 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 modules, direct sum of two non-zero submodules. Every simple module is indecomposable, but there are indecomposable modules which are not simple (e.g. uniform modules). ; 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 (ring theory), annihilator of ''M'' is the zero ideal. ; Torsion-free: A torsion-free module is a module over a ring such that 0 is the only element annihilated by a regular element (non zero-divisor) of the ring, equivalently $rm=0$ implies $r=0$ or $m=0$. ; Noetherian: A Noetherian module is a module which satisfies the ascending chain condition 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 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 homomorphism, group endomorphism of the abelian group . The set of all group endomorphisms of ''M'' is denoted EndZ(''M'') and forms a ring under addition and function composition, composition, and sending a ring element ''r'' of ''R'' to its action actually defines a ring homomorphism from ''R'' to EndZ(''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 on . 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
integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of t ...
s or over some modular arithmetic Z/''n''Z.

## Generalizations

A ring ''R'' corresponds to a preadditive category R with a single object (category theory), object. With this understanding, a left ''R''-module is just a covariant additive functor from R to the category of abelian groups, 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 (''X'', O''X'') and consider the sheaf (mathematics), sheaves of O''X''-modules (see sheaf of modules). These form a category O''X''-Mod, and play an important role in modern
algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

. 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 monoids. 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
incorporating the semirings from theoretical computer science. Over near-rings, one can consider near-ring modules, a nonabelian generalization of modules.

* Group ring * Algebra (ring theory) * Module (model theory) * Module spectrum *Annihilator (ring theory), Annihilator

# 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,