Presentation of finite rings (fields) One knows that every finite group is isomorphic to a subgroup of $\operatorname{GL}(n)$ for some $n$ large enough. 

Can every finite ring be represented by a ring of matrices, i.e., is every ring isomorphic to a matrix ring?

As an example, let $G = S_3$, the symmetric group on $3$ points. Let $g_1 = (1,2,3)$ and $g_2 = (1,2)$ the generator of $G$. Then the matrix group with gnerators
$$
M_1 = \begin{bmatrix}
0 & 0 & 1 \\
1 & 0 & 0 \\
0 & 1 & 0 \end{bmatrix}$$
$$
M_2 = \begin{bmatrix}
0 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 1 \end{bmatrix}$$
is isomorphic to $G$.

At least the following is true: If $R$ is some $k$-algebra, then it embeds into $\mathrm{End}_k(U(R))$, where $U(R)$ is the $k$-module underlying $R$. The embedding is just $r \mapsto (m \mapsto rm)$, and can also be seen as a special case of the Yoneda embedding. When $k$ is a field, this module is free, hence the endomorphism ring is a ring of (column-finite) matrices. In particular, every finite-dimensional algebra over a field embeds into some finite-dimensional matrix algebra.
