Annihilator of maximal ideals in a finite dimensional algebra

Question

I wonder if the following is correct:

The left (right) annihilator of every (2 sided) maximal ideal in a finite dimensional $k$-algebra is always nonzero.

Clearly this is true for semi-simple algebras. Is this true for any finite dimensional algebra? If not, please provide a counterexample.

Answer 1

I think I found a counterexample, very simple: the $2 \times 2$ upper triangular matrix over a field $k$. The subspace $$\begin{pmatrix} 0 & k \\ 0 & k\end{pmatrix}$$ is a two sided maximal ideal. But its left annihilator is zero.