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 $G$.

Not every group is isomorphic to a matrix group. See for example this MO post for counterexamples. It is true, however, that every finite group is isomorphic to a matrix group over arbitrary field.
If you meant to ask if every finite ring is isomorphic to a matrix ring over some field, then this is not true: a matrix ring over $k$ is a $k$-algebra, in particular its characteristic is equal to that of $k$, so any ring with nonprime characteristic (like ${\bf Z}/6{\bf Z}$) can't be represented as a matrix ring over a field.
On the other hand, every ring is trivially isomorphic to the ring of $1\times 1$ matrices over itself, or more generally $n\times n$ scalar matrices over itself.
