In

^{''p''} spaces.)

_{''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

module">'x''mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, 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 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 g ...

, 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 branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ...

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 that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...

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 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 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
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...

.
Introduction and definition

Motivation

In a vector space, the set of scalars is afield
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 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 gen ...

. 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 Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

. However, modules can be quite a bit more complicated than vector spaces; for instance, not all modules have a basis
Basis may refer to:
Finance and accounting
*Adjusted basisIn tax accounting, adjusted basis is the net cost of an asset after adjusting for various tax-related items.
Adjusted Basis or Adjusted Tax Basis refers to the original cost or other b ...

, and even those that do, free module
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

s, need not have a unique rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking
A ranking is a relationship between a set of items such that, for any two items, the first is either "rank ...

if the underlying ring does not satisfy the invariant basis number
In mathematics, more specifically in the field of ring theory, a ring (mathematics), ring has the invariant basis number (IBN) property if all finitely generated free module, free left module (mathematics), modules over ''R'' have a well-defined ran ...

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 mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

in general, but not in the case of finite-dimensional spaces, or certain well-behaved infinite-dimensional spaces such as LFormal definition

Suppose that ''R'' is a ring, and 1 is its multiplicative identity. A left ''R''-module ''M'' consists of anabelian group
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 g ...

and an operation such that for all ''r'', ''s'' in ''R'' and ''x'', ''y'' in ''M'', we have
#$r\; \backslash cdot\; (\; x\; +\; y\; )\; =\; r\; \backslash cdot\; x\; +\; r\; \backslash cdot\; y$
#$(\; r\; +\; s\; )\; \backslash cdot\; x\; =\; r\; \backslash cdot\; x\; +\; s\; \backslash cdot\; x$
#$(\; r\; s\; )\; \backslash cdot\; x\; =\; r\; \backslash cdot\; (\; s\; \backslash cdot\; x\; )$
#$1\; \backslash cdot\; x\; =\; x\; .$
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 glossary of ring theory
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 studies th ...

, all rings and modules are assumed to be unital.
An ''(R,S)''-bimoduleIn abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ri ...

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 $(r\; \backslash cdot\; x)\; \backslash ast\; s\; =\; r\; \backslash cdot\; (\; x\; \backslash ast\; s\; )$ for all ''r'' in ''R'', ''x'' in ''M'', and ''s'' in ''S''.
If ''R'' is commutative
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

, then left ''R''-modules are the same as right ''R''-modules and are simply called ''R''-modules.
Examples

*If ''K'' is afield
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
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). ...

, then a module.html" ;"title="'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
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

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
Rationality is the quality or state of being rational – that is, being based on or agreeable to reason
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: grc, wikt:λογι ...

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
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 g ...

is a module over the ring of integer
An integer (from the Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ...

s Z in a unique way. For , let (''n'' summands), , and . Such a module need not have a basis
Basis may refer to:
Finance and accounting
*Adjusted basisIn tax accounting, adjusted basis is the net cost of an asset after adjusting for various tax-related items.
Adjusted Basis or Adjusted Tax Basis refers to the original cost or other b ...

—groups containing torsion elements do not. (For example, in the group of integers 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
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 ...

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
The decimal numeral system (also called the base-ten positional numeral system, and occasionally called denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the H ...

(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 total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...

, then the cartesian product
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

''R''free
Free may refer to:
Concept
* Freedom, having the ability to act or change without constraint
* Emancipate, to procure political rights, as for a disenfranchised group
* Free will, control exercised by rational agents over their actions and decis ...

and if ''R'' has invariant basis number
In mathematics, more specifically in the field of ring theory, a ring (mathematics), ring has the invariant basis number (IBN) property if all finitely generated free module, free left module (mathematics), modules over ''R'' have a well-defined ran ...

(e.g. any commutative ring or field) the number ''n'' is then the rank of the free module.
*If Mmatrices
Matrix or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics)
In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of numbers, symbol (formal), symbols, or expression (mathema ...

over a ring ''R'', ''M'' is an Mcategory
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization
Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...

of Mequivalent
Equivalence or Equivalent may refer to:
Arts and entertainment
*Album-equivalent unit, a measurement unit in the music industry
*Equivalence class (music)
*''Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre
*''Equivalent ...

. The special case is that the module ''M'' is just ''R'' as a module over itself, then ''R''nonempty
In mathematics, the empty set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exists by includ ...

set, ''M'' is a left ''R''-module, and ''M''function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

s , then with addition and scalar multiplication in ''M''smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold
The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's s ...

, then the smooth function
In mathematical analysis
Analysis is the branch of mathematics dealing with Limit (mathematics), limits
and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mat ...

s from ''X'' to the real number
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 g ...

s form a ring ''C''vector field
In vector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the process of making a product different from other similar products
* Product ...

s defined on ''X'' form a module over ''C''tensor field
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

s and the differential form
In the mathematical
Mathematics (from Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population ...

s on ''X''. More generally, the sections of any vector bundle
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 gen ...

form a projective module
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 gener ...

over ''C''category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization
Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...

of ''C''equivalent
Equivalence or Equivalent may refer to:
Arts and entertainment
*Album-equivalent unit, a measurement unit in the music industry
*Equivalence class (music)
*''Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre
*''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 In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...

''R''underlying set
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

and the same addition operation, but the opposite multiplication: if in ''R'', then in ''R''universal enveloping algebra
In mathematics, a universal enveloping algebra is the most general (unital algebra, unital, associative algebra, associative) algebra that contains all representation of a Lie algebra, representations of a Lie algebra.
Universal enveloping algebras ...

.
*If ''R'' and ''S'' are rings with a ring homomorphism
In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function (mathematics), function between two ring (algebra), rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function s ...

''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 asubgroup
In group theory, a branch of mathematics, given a group (mathematics), group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely ...

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
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

of an ''R''-module, then the submodule spanned by ''X'' is defined to be $\backslash langle\; X\; \backslash rangle\; =\; \backslash ,\backslash bigcap\_\; N$ where ''N'' runs over the submodules of ''M'' which contain ''X'', or explicitly $\backslash left\backslash $, 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:
Given submodules ''U'', ''N''map
A map is a symbol
A symbol is a mark, sign, or that indicates, signifies, or is understood as representing an , , or . Symbols allow people to go beyond what is n or seen by creating linkages between otherwise very different s and s. A ...

is a homomorphism of ''R''-modules if for any ''m'', ''n'' in ''M'' and ''r'', ''s'' in ''R'',
:$f(r\; \backslash cdot\; m\; +\; s\; \backslash cdot\; n)\; =\; r\; \backslash cdot\; f(m)\; +\; s\; \backslash cdot\; f(n)$.
This, like any homomorphism
In 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 mathematical analysis, analysis. I ...

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
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

.
A bijective
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

module homomorphism is called a module isomorphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

, 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
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learnin ...

of a module homomorphism is the submodule of ''M'' consisting of all elements that are sent to zero by ''f'', and the image
An image (from la, imago) is an artifact that depicts visual perception
Visual perception is the ability to interpret the surrounding environment (biophysical), environment through photopic vision (daytime vision), color vision, sco ...

of ''f'' is the submodule of ''N'' consisting of values ''f''(''m'') for all elements ''m'' of ''M''. The isomorphism theorem
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 ...

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
In mathematics, an abelian category is a Category (mathematics), category in which morphisms and Object (category theory), objects can be added and in which Kernel (category theory), kernels and cokernels exist and have desirable properties. The mo ...

, denoted by ''R''-Mod (see category of modulesIn 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 mathematical analysis, analysis. In i ...

).
Types of modules

; Finitely generated: An ''R''-module ''M'' is finitely generated if there exist finitely many elements ''x''linear combination
In mathematics, a linear combination is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' and ''y'' would be ...

of those elements with coefficients from the ring ''R''.
; Cyclic: A module is called a cyclic module In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...

if it is generated by one element.
; Free: A is a module that has a basis, or equivalently, one that is isomorphic to a direct sum
The direct sum is an operation from abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathema ...

of copies of the ring ''R''. These are the modules that behave very much like vector spaces.
; Projective: Projective module
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 gener ...

s are direct summand
The direct sum is an operation from abstract algebra, a branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geome ...

s of free modules and share many of their desirable properties.
; Injective: Injective module
In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module (mathematics), module ''Q'' that shares certain desirable properties with the Z-module Q of all rational numbers. Specifically, if ''Q ...

s are defined dually to projective modules.
; Flat: A module is called flat if taking the tensor product
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

of it with any exact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups
A group is a number of people or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same ...

of ''R''-modules preserves exactness.
; Torsionless: A module is called torsionless if it embeds into its algebraic dual.
; Simple: A simple moduleIn mathematics, specifically in ring theory, the simple modules over a Ring (mathematics), ring ''R'' are the (left or right) module (mathematics), modules over ''R'' that are Zero_element#Zero_module, non-zero and have no non-zero proper submodules. ...

''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 (mathematics), ring that is a semisimple ...

is a direct sum (finite or not) of simple modules. Historically these modules are also called ''completely reducible''.
; Indecomposable: An indecomposable moduleIn abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ri ...

is a non-zero module that cannot be written as a direct sum
The direct sum is an operation from abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathema ...

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
In mathematics, the annihilator of a subset of a Module (mathematics), module over a ring (mathematics), ring is the ideal (ring theory), ideal formed by the elements of the ring that give always zero when multiplied by an element of .
Over an ...

''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
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 ...

.
; Torsion-free: A torsion-free module
In abstract algebra, algebra, a torsion-free module is a module (mathematics), module over a Ring (mathematics), ring such that zero is the only element Absorbing element, annihilated by a zero-divisor, regular element (non zero-divisor) of the ring ...

is a module over a ring such that 0 is the only element annihilated by a regular element (non zero-divisor
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...

) of the ring, equivalently $rm=0$ implies $r=0$ or $m=0$.
; Noetherian: A Noetherian moduleIn abstract algebra, a Noetherian module is a module that satisfies the ascending chain condition on its submodule
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mat ...

is a module which satisfies the ascending chain conditionIn mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly Ideal (ring theory), ideals in certain commutative rings.Jacobson (2009), p. 1 ...

on submodules, that is, every increasing chain of submodules becomes stationary after finitely many steps. Equivalently, every submodule is finitely generated.
; Artinian: An Artinian moduleIn abstract algebra, an Artinian module is a module (mathematics), module that satisfies the descending chain condition on its poset of submodules. They are for modules what Artinian rings are for rings, and a ring is Artinian if and only if it is an ...

is a module which satisfies the descending chain conditionIn 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 ha ...

on submodules, that is, every decreasing chain of submodules becomes stationary after finitely many steps.
; Graded: A graded module
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 ...

is a module with a decomposition as a direct sum over a graded ring
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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 thegroup ring
In 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 mathematical analysis, analysis. I ...

''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 of the abelian group . The set of all group endomorphisms of ''M'' is denoted Endcomposition
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, and sending a ring element ''r'' of ''R'' to its action actually defines a ring homomorphism
In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function (mathematics), function between two ring (algebra), rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function s ...

from ''R'' to Endinjective
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

. 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
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s or over some modular arithmetic #REDIRECT Modular arithmetic #REDIRECT Modular arithmetic#REDIRECT Modular arithmetic
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure ( ...

Z/''n''Z.
Generalizations

A ring ''R'' corresponds to apreadditive category In mathematics, specifically in category theory, a preadditive category is
another name for an Ab-category, i.e., a category (mathematics), category that is enriched category, enriched over the category of abelian groups, Ab.
That is, an Ab-catego ...

R with a single object
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. With this understanding, a left ''R''-module is just a covariant additive functorIn mathematics, specifically in category theory, a preadditive category is
another name for an Ab-category, i.e., a category (mathematics), category that is enriched category, enriched over the category of abelian groups, Ab.
That is, an Ab-categor ...

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 categoryIn category theory
Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled dire ...

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 spaceIn mathematics
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(''X'', Oalgebraic geometry
Algebraic geometry is a branch of mathematics
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. If ''X'' has only a single point, then this is a module category in the old sense over the commutative ring Osemiring
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. Modules over rings are abelian groups, but modules over semirings are only commutative
In mathematics
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monoid
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...

s. Most applications of modules are still possible. In particular, for any semiring
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...

''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.
See also

*Group ring
In 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 mathematical analysis, analysis. I ...

* Algebra (ring theory)
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 ...

* 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
Nathan Jacobson (October 5, 1910 – December 5, 1999) was an American
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. ''Structure of rings''. Colloquium publications, Vol. 37, 2nd Ed., AMS Bookstore, 1964,
External links

*Why is it a good idea to study the modules of a ring ?

on

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*
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Algebraic structures
* Module