All finite-dimensional simple modules are $1$-dimensional

Question

Let $A$ be a (non-commutative) $k$-algebra, where $k$ is an algebraically closed, characteristic zero field. Let $M$ be a finite-dimensional simple $A$-module. If $A/\operatorname{ann}(M)$ is commutative, then $M$ is $1$-dimensional. Is there a characterization (or at the very least, a name) for algebras in which this holds for all finite-dimensional simple modules? A simple example would be the quantum plane $\mathcal{O}_q(k^2)$ at $q \in k^\times$ a non-root of unity.