Module (mathematics)

A module [ mo ː module ] ( masculine, plural: moduli [ mo ː moduli ] ) is an algebraic structure that represents a generalization of a vector space.

• 2.1 Alternative definitions
• 2.2 bimodules

Modules over a commutative ring with unit element

In this simplest case, one can directly write down the axioms of a vector space and everywhere " body " with " ring " Replace: A module over a commutative ring with unit element is an abelian group together with a picture

Such that:

You Prompts additionally, it is called the unitary module.

The study of these modules is subject of commutative algebra.

Abelian groups

Every abelian group is uniquely a unitary module: Due

Are at most

And analogously

Conceivable ( for natural numbers ). But since this only possible link satisfies the module axioms, the assertion follows. ( Here, the abelian group was written additively ).

Vector spaces with a linear mapping in itself

Be the polynomial ring over a field. Then the moduli correspond one -to-one pairs consisting of a vector space and an endomorphism of:

• Be a module. We note that a vector space is, there is embedded in. Let this vector space. The matched pair is now to where by

• For a pair, we define a module structure by

We put

Ring ideals

Each ring is a module over itself with the ring multiplication as the operation. The submodules then correspond exactly to the ideals of ( as in this section is commutative, we need not between left and right ideals to distinguish ).

Modules over an arbitrary ring

It was a ring. Is not commutative, one has to distinguish between left and right modules.

A - Links module is an abelian group together with a picture

Which is additive in both arguments, that is, for all valid

And for

Applies. It is assumed that a unitary ring, as is demanded in most cases that the links module is unitary, i.e.,

A -module is an abelian group together with an additive in both arguments illustration

So that

A right module over a unitary ring is unitary if

Is commutative, then agree the terms left and right module (except for the spelling), respectively, and one simply speaks of -modules.

Alternative definitions

• A - Links module is an abelian group together with a (possibly unitary ) ring homomorphism

• A -module is an abelian group together with a (possibly unitary ) ring homomorphism

Bimodules

Let and rings. Bimodule is an abelian group together with a links module and a right - module structure so that - then one is

Applies.

Alternatively, one is - bimodule is an abelian group together with a ring homomorphism

Modules over an associative algebra

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

So that

Applies.

A -module is a module together with a - Modulhomomorphismus

So that

Applies.

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

Modules over a Lie algebra

It is a Lie algebra over a field. A module or a representation of a vector space together with a bilinear map -

So that

Applies.

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

Here is the algebra of endomorphisms of the commutator as Lieklammer.

-Modules are the same as universal enveloping algebra of the moduli of.

Modules over a group

It is a group. A module or accurate links module is an abelian group together with an external two -digit shortcut

So that

And

As well as

Applies.

A -module is defined analogously; the second condition is characterized by

To replace.

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

While the group of automorphisms of is with linking

A -module is an abelian group together with a group homomorphism

The product to be carried

Given.

: Module is an abelian group with a module and a module structure that are compatible in the following sense - Is on a ring, then a is

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

This is the group of automorphisms of a module.

- Modules are the same as modules over the group ring.

Specifically, is a body, then the concept of true - module with that of the linear representation of the same.