The category of algebras over a monad over a category $\mathcal{A}$ is regular, if $\mathcal{A}$ is. A monad is a monoid in the monoidal category $(\mathcal{A}^\mathcal{A},\circ,\dots)$. If $\mathcal{A}$ is regular, then so is $\mathcal{A}^\mathcal{A}$.
This suggests that there is a reasonable condition we can put on a monoidal category $\mathcal{X}$ such that for every monoid $M$ in $\mathcal{X}$ the category of modules over $M$ is regular.
Is this a reasonable hypothesis and is anything known about this idea?