When is the subring (containing 1) of a matrix ring $M_n(k)$ over a field $k$ is local? I would be grateful for every reference concerning this matter, Thank you!

The ring of matrices $ \left( \begin{array}{cc} a & b \\\\ 0 & a \\ \end{array} \right). $ This ring is isomorphic to the algebra of dual numbers (http://en.wikipedia.org/wiki/Dual_number) which is local. 


The idea of @borisnovikov shows that the algebra of all upper triangular matrices such that the entries of the main diagonal are equal, is a local ring. The dimension of this algebra is $\frac{n^2n}{2}+1$. I am wondering if one can find a local subalgebra of $M_n(k)$ whose dimension is greater than this number. 

