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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.

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    $\begingroup$ ${\mathbb R}(x)\not\subset C^\infty({\mathbb R})$. $\endgroup$ Feb 17, 2014 at 13:13
  • $\begingroup$ @Sasha Anan'in: Thanks a lot. I have edited the question to make things (hopefully :) ) correct. $\endgroup$ Feb 17, 2014 at 14:27
  • $\begingroup$ Also, just to clarify: I would be particularly interested in the examples that are much smaller than those listed at the end of the question. $\endgroup$ Feb 17, 2014 at 15:58
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    $\begingroup$ There are no smaller ones because ${\mathbb C}[z]$ (and similarly ${\mathbb R}[x]$) is a simple module over the Weyl ${\mathbb C}$-algebra. $\endgroup$ Feb 17, 2014 at 16:57
  • $\begingroup$ @Sasha Anan'in: Thanks, must ponder over that. $\endgroup$ Feb 17, 2014 at 17:03

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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.

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  • $\begingroup$ I have some doubts that $D$-subrings form "essentially the set of $D$-stable ideals of $C^\infty({\mathbb R})$". And a wider class can easily admit a handy description. (Not that I claim this is the case here.) $\endgroup$ Feb 17, 2014 at 13:29
  • $\begingroup$ Absolutely, there are (probably) more nonunital subrings that are $D$-modules than just the $D$-stable ideals, I'll edit in a clarification. $\endgroup$ Feb 17, 2014 at 13:58
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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 $0$.

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