 Right Artinian triangular ring that is not left Noetherian

Let $K/k$ be an infinite-dimensional field extension. For the triangular ring in the display, the right regular module has length three: $e_{1}T$ has its off-diagonal simple submodule and simple quotient, and $e_{2}T$ is simple. Thus $T$ is right Artinian. Its off-diagonal ideal is, on the left, the infinite-dimensional $k$-vector space $K$. Every $k$-subspace determines a left ideal, so an ascending sequence of finite-dimensional subspaces violates left Noetherianity. For example take $K=k(z)$ and the spans of $1,z,\dots ,z^m$.