The Weyl algebra modules which are also rings 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.
 A: Recall that an ideal is in particular a non-unital subring; so a subclass of the non-unital part of what you are asking for is the set of $D$-stable ideals of $C^\infty(\mathbb{R})$, i.e. those ideals that are also $D$-modules. This is the intersection of two rather hard problems; a) finding/describing general ideals of $C^\infty(\mathbb{R})$ and b) describing the $D$-stable ideals of a ring that is a $D$-module.
For unital subrings that are also $D$-modules, I suppose any subring you could recognise as a ring of functions will do, e.g. polynomials or real-analytic functions $C^\omega(\mathbb{R})$ (note that the ring of rational functions is not a subring of the ring of smooth functions), but that is of course a rather coarse description given that e.g. there are certainly very many rings lying between $C^\omega(\mathbb{R})$ and $C^\infty(\mathbb{R})$ that will work.
A: A natural class of examples is given by localizations of $\Bbb{C}[z]$, or in a more geometric language, rational functions with prescribed singularities. For example, Laurent polynomials $\Bbb{C}[z,z^{-1}]$ correspond to localizing (i.e. allowing poles) at $0$.  
