Classification of two dimensional algebras I am interested in find up to isomorphisms all two dimensional algebras (associative and with unit) over a field $K$.
I have proceeded as follows:
I know that every finite dimensional algebra is isomorphic to an subalgebra of $M_n(K)$.
So, I have looked for all subalgebras of $M_2(K)$ and I have found \begin{bmatrix}{K}&{0}\\{0}&{K}\end{bmatrix},\begin{bmatrix}{K}&{0}\\{K}&{K}\end{bmatrix} and \begin{bmatrix}{K}&{K}\\{0}&{K}\end{bmatrix}
Are they all two dimensional $K$-algebras?
 A: Note that the upper and lower triangular matrix subalgebras you have chosen are obviously $3$ dimensional. 
Doing the classification by searching for all subalgebras is going to be a rough go, especially considering you may have trouble seeing which ones are mutually isomorphic. I'd suggest realizing them as quotients of a polynomial ring: in this case, $K[X]$ suffices.
We can pick $1$ as a basis element, and then select something linearly independent, say $x$.  Map from $K[X]$ onto $A$ with the rule $k\mapsto k$ for all $k\in K\subseteq K[X]$, and $X\mapsto x$. This uniquely determines an algebra homomorphism to $A$. It is necessarily surjective since the image is at least two dimensional. By the first isomorphism theorem $K[X]/(f)\cong A$ for some quadratic monic polynomial $f$.
There are three possibilities: 


*

*$f$ is irreducible

*$f$ has two distinct linear factors

*$f$ has one linear factor of multiplicity 2


I leave it to you to describe these resulting rings in plain terms.
Note that the first case can’t happen when $K$ is quadratically closed , but the other two cases happen for any field.

It would be a good exercise to now go back and demonstrate some examples realized as $2\times 2$ matrices!  I will list a set and you can work out which is which
$\left\{\begin{bmatrix}a&0 \\ 0 &b\end{bmatrix}\,\middle|\,a,b\in\mathbb R\right\}$. 
$\left\{\begin{bmatrix}a&b \\ -b &a\end{bmatrix}\,\middle|\,a,b\in\mathbb R\right\}$
$\left\{\begin{bmatrix}a&b \\ 0 &a\end{bmatrix}\,\middle|\,a,b\in\mathbb R\right\}$.
