Suppose that $k$ is a field and that $R$ is a semisimple ring that is also a $k$-algebra with $\dim_k R$ at most $3$. Show that $R$ is commutative and that the semisimplicity is required.
The case when $R$ has dimension $1$ is simple as it is routine to verify that then $R \cong k$. As for the remaining two dimensions, I am unsure of how to continue. It is, however, clear to me that this must make use of the Artin-Wedderburn Theorem and the reason this fails for higher $n$ is that the larger the $n$, the 'more' non-commutative the matrix rings become. I assume the conditions on $R$ give just enough structure to force commutativity in the small matrix rings. How do I go about proving the result for $\dim_k R=2$ and $\dim_k R=3$?