## Algebras

It’s been a long time since my last post about mathematics, which is kind of a shame, because I wanted math to be one of the main subjects of this blog.

So, I decided to start and write a little bit more about things I loved in mathematics (and hopefully I will follow).

In this post, I’m assuming some basic Linear Algebra knowledge. This post is a little boring, but it defines an important algebraic structure that is used in some very beautiful subjects and theorems, so stay tuned.

### Algebras and Subalgebras

Definition:Analgebrais a vector space over field , endowed with a binary bilinear operation () s.t. :

For example, The polynomials with variables, , with the multiplication as the operation is an associative and commutative algebra. This algebra is usually called the Polynomial algebra.

But please note, **NOT** all algebras must be commutative! For example, we can look at the vector space of the matrices (), with matrix multiplication at the operation . is an associative, non-commutative algebra.

We can also define a subalgebra:

Definition:A subspace is called asubalgebraif .

For example, the diagonal matrices are a subalgebra of the matrices algebra.

#### Homomorphism

And just like any other algebraic structure, we can define an *homomorphism* between two algebras as a linear map between the algebras the preserve the operation, that is:

Defenition:Let be algebras, a linear map is called an homomorphism if:

### Ideals

Definition:A subspace of an algebra is aleft(resp.right, resp.two-sided)idealif:

(resp. , resp. ).

It is clear, by the definition, that any ideal is a subalgebra.

Using the ideals, we can look at the quotient space (a space where any vector in I means the zero vector, meaning that two vector that differ by only a vector in I are identical). The quotient space has the canonical algebra structure given by:

which called the quotient algebra .

It’s easy to see that the canonical map that is given by is an algebra homomorphism.

Moreover, if is an algebra homomorphism, the kernel is a two-sided ideal of and the image is a subalgebra of .