Module (mathematics)

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A module [ moːdʊl ] (masculine, plural modules [ moːdʊln ], the declination is similar to that of consul ; of Latin modulus , diminutive of mode , "degree", "unit") is an algebraic structure that a generalization of a vector space represents.

Similar to rings , a module is understood to mean something different depending on the sub-area and textbook. The definitions of morphisms as well as substructures and superstructures are also slightly different. In mathematical terms, these different module terms are different categories .

Modules over a commutative ring with a single element

A module over a commutative ring or module for short is an additive Abelian group together with a mapping

          (called multiplication by scalars, scalar multiplication),

so that:

If one requires additionally for an identity with


so one calls the module unitary (English: unital ). Some authors generally require the existence of a unit element for rings, and then likewise for modules over rings. If there is a body , i.e. also forms an Abelian group, then the unitary modules are over just the vector spaces over .

Comment: The concept of vector space is actually superfluous, since it is a special case of the more general concept of unitary module. In fact, however, the additional condition that there is a body enables so many results that are not correct in the general situation that it is customary to separate the special case from the general case with a separate term.

The study of modules on commutative rings is the subject of commutative algebra .

Abelian groups

Each additive Abelian group is uniquely a unitary module; H. a unitary module over the commutative ring of integers. Be . Because of

must apply to with :

and analog:


Since this only possible connection fulfills the modulus axioms, the claim follows. The following number ranges are additive groups and thus modules:

Upper rings as modules

Is a top ring of , it is by definition an Abelian group.

If the ring multiplication of is restricted to the set , then this defines the necessary scalar multiplication in order to consider it as a module over in a natural way . If and if they have the same single element, the module is unitary.

If and even bodies are, one speaks in this situation of a body expansion . The module structure then becomes a vector space structure as described above. The consideration of this vector space structure is an indispensable aid in the investigation of body extensions.

Note: The number ranges mentioned in the previous chapter are all upper rings of , which also shows that they are naturally modules.

Vector spaces with a linear map in themselves

Let be the polynomial ring over a body . Then, the corresponding -modules one-to-one pairs consisting of a vector space and an endomorphism on .

  • Be a module. We note that a vector space is as in embedded. Let this vector space. The pair that belongs to it is now , being through
given is.
  • We define a module structure for a pair
and continue the -linear on , d. H. for all
let's set

Ring ideals

Each ring is a module on itself with ring multiplication as an operation. The sub-modules then correspond exactly to the ideals of (since in this section is commutative, we do not need to distinguish between left and right ideals).

Modules over any ring

Let it be a ring . If this ring is not (necessarily) commutative, a distinction must be made between left and right modules.

A - left module is an Abelian group along with a ring and a map

which is additive in both arguments, d. H. for all true

  • and

and for the

  • for all


If it is assumed that there is a unitary ring with a one-element , then it is usually also required that the left module is unitary (English: unital ), i.e. H.

  • for everyone .

Some authors generally require the existence of a single element for rings and modules.

A right module is defined similarly, except that the scalars of the ring act from the right on the elements of : A - right module is an Abelian group together with a mapping that is additive in both arguments

so that

for all

A right module over a unitary ring with one element is unitary if

applies to all .

If commutative , the terms left and right module match (except for the spelling), and one speaks simply of - modules . Usually the above notation is used for left modules.

Alternative definitions

  • A left module is an Abelian group together with a (possibly unitary) ring homomorphism
It is the ring of endomorphism of the concatenation as a product:
  • A right module is an Abelian group together with a (possibly unitary) ring homomorphism
It should be the counter-ring of endomorphism ring, that is the ring of endomorphisms of the legal chaining as a product:


Let it be and rings. Then a - -Bimodule is an Abelian group together with a -Left module- and a -Right module structure, so that



For unitary rings and a unitary - -Bimodule (i.e. with for all ) can alternatively be described as an Abelian group together with a unitary ring homomorphism

That means: A unitary - -Bimodule is nothing more than a unitary -Link module.

Change of ring

and be rings and be a ring homomorphism. The regulation defines for each module

a module structure called the one associated with and the module structure . This module is labeled with or with . If in particular a subring of and is the canonical embedding, then the module obtained by restricting the scalars from to is called.

If a sub-module is of , then is a sub-module of and

Modules over an associative algebra

If a commutative ring and an associative R -algebra , then a - left module is a module together with a module homomorphism

so that



A - Law Module is a module together with a -Modulhomomorphismus

so that



Unitary modules and bimodules are defined analogously to the case of rings.

Modules over a Lie algebra

Let it be a Lie algebra over a field . A - module or a representation of a vector space together with a -bilinearen Figure

so that



Alternatively, a module is a vector space along with a homomorphism of Lie algebras over

where is the -algebra of the endomorphisms of with the commutator as a Lie bracket.

-Modules are the same as modules under the universal enveloping algebra of .

Modules over a group

It is a group . A module or, more precisely, left module is an Abelian group together with an outer two-digit link


so that

for all


for all

such as

for the neutral element of and for everyone


A - law module is defined similarly; the second condition is through

for all

to replace.

Alternatively, a - (left) module is an Abelian group together with a group homomorphism

here is the group of automorphisms from with the link


A right module is an Abelian group together with a group homomorphism

the product on is through



If further is a ring, then a - module is an Abelian group with a module structure and a module structure which are compatible in the following sense:


Alternatively, a - module is a module together with a group homomorphism

this is the group of automorphisms of the module.

- modules are the same as modules above the group ring .

If a body is special , the concept of the - module corresponds to that of the linear representation of .

See also

Web links

Wiktionary: Module  - explanations of meanings, word origins, synonyms, translations

Alexander von Felbert: Introduction to module theory .


References and comments

  1. not to be confused with scalar product
  2. a b David S. Dummit, Richard M. Foote: Abstract Algebra . John Wiley & Sons, Inc., Hoboken, NJ 2004, ISBN 978-0-471-43334-7 .
  3. Here the Abelian group was written additively.
  4. Such a module does not have to have a basis , namely for modules with torsion elements .
  5. ^ Nicolas Bourbaki: Elements of Mathematics, Algebra I, Chapters 1–3 . 2nd Edition. Springer, 1998, ISBN 3-540-64243-9 , § 3. Tensor products , 2., p. 221 ( Internet Archive ).