The $D$-module structure of $R[F]$-modules

Let $R$ be a regular ring, essentially of finite type over a perfect field $k$. An $R$-module $\mathcal{M}$ is called a unit $R[F]$-module if it comes equipped with an isomorphism $F^{e*} \mathcal{M} \xrightarrow{ \ }\mathcal{M}$, where $F$ denotes the Frobenius map on $\operatorname{Spec}R$, and $F^{e*}$ is the associated pullback functor. It is well known that $\mathcal{M}$ then carries a natural $D_R$-module structure. In this paper we investigate the relation between the unit $R[F]$-structure and the induced $D_R$-structure on $\mathcal{M}$. In particular, it is shown that if $k$ is algebraically closed and $\mathcal{M}$ is a simple finitely generated unit $R[F]$-module, then it is also simple as a $D_R$-module. An example showing the necessity of $k$ being algebraically closed is also given.