Let $(R,\mathfrak{m})$ be a local ring over a field. Suppose the ring has flat functions, i.e. $\mathfrak{m}^\infty\neq\{0\}$. (The prototype is of course $C^\infty(\Bbb{R}^p,0)$, or a quotient of it, by some finitely generated ideal.)
Is there a notion of the 'infinite-radical' of an ideal, e.g. $\sqrt[\infty]{\mathfrak{m}^\infty}$ ? I would like in the 'geometric' case $\sqrt[\infty]{\mathfrak{m}^\infty}$ to be the defining ideal of the set $V(\mathfrak{m}^\infty)$. For example, for the ring $R=k[[\underline{y}]]\otimes C^{\infty}(\Bbb{R}^p,0)$ we have $\mathfrak{m}=(\underline{x},\underline{y})$, here $\underline{x}$ are the local coordinates on $(\Bbb{R}^p,0)$. Therefore $\mathfrak{m}^\infty=(\underline{x})^\infty$. Thus $\sqrt[\infty]{\mathfrak{m}^\infty}=(\underline{x})\neq\mathfrak{m}$.
What is the definition of the height of $\mathfrak{m}^\infty$? For some rings $\mathfrak{m}^\infty$ is a prime ideal, (e.g. if the $\mathfrak{m}$-adic completion is a domain), is there some general characterization? Any general results about the prime sub-ideals $\mathfrak{p}\subset\mathfrak{m}^\infty$? At least for the case of $C^\infty(\Bbb{R}^p,0)$?
Is there any place summarizing the known results about rings with the 'flat functions'? (Maybe extending some properties of flat elements of $C^\infty(\Bbb{R}^p,0)$ to more general rings? Some analogue of Tougeron's book for more general rings?)
I have asked this on matheoverflow, no replies :(