Consider the space $C^\infty(\mathbb{R})$. We have on it a natural action of the Weyl algebra generated by $x$ and $d/dx$, where $x$ is the coordinate on $\mathbb{R}$, and there is plenty of Weyl algebra modules $V\subset C^\infty(\mathbb{R})$. Some of those modules are also rings, and I would like to know as many such examples as possible.
EDIT: After Sasha Anan'in comment I realized that I am also interested (perhaps even more than in the original case) in the situation when $\mathbb{R}$ is replaced by $\mathbb{C}$, so instead of $x\in\mathbb{R}$ and $d/dx$ we have $z\in\mathbb{C}$ and $d/dz$ now, and $C^\infty(\mathbb{R})$ is replaced by the space of meromorphic functions $M(\mathbb{C})$ on $\mathbb{C}$.
Question: is there a complete description, or at least some broad classes of examples, of the Weyl algebra modules $V\subset M(\mathbb{C})$ (or $V\subset C^\infty(\mathbb{R})$) being simultaneously rings, not necessarily with unity?
Simple examples of such modules are provided by the spaces $\mathbb{C}[z]$ and $\mathbb{R}[x]$ of polynomials in $z$ and $x$, and $\mathbb{C}(z)$ of rational functions in $z$, but I hope that there should be much more.