rings with 'flat functions'

Let $(R,\mathfrak{m})$ be a local ring over a field. Suppose the ring has flat elements, 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.)

1. For which rings and ideals, $J\subset R$, the following holds. If the completions satisfy $\hat{J}\supseteq(\widehat{\mathfrak{m}})^N$ then $J\supseteq\mathfrak{m}^{N+n}$, for some finite $n$. At least $J\supseteq\mathfrak{m}^{\infty}$? (Probably a better/more important version: suppose the image of $J$, under the completion map $R\rightarrow\hat{R}$, contains $(\widehat{\mathfrak{m}})^N$. Do we have $J\supseteq\mathfrak{m}^{N}$? This seems to hold when $\mathfrak{m}^N$ is finitely generated)

1. Is there a notion of the 'infinite-radical' of an ideal, $\sqrt[\infty]{\mathfrak{m}^\infty}$ ? 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 $\mathfrak{m}^\infty=(\underline{x})^\infty$, here $\underline{x}$ are the local coordinates on $(\Bbb{R}^p,0)$. Thus $\sqrt[\infty]{\mathfrak{m}^\infty}=(\underline{x})$.

(upd: What is the definition of the height of $\mathfrak{m}^\infty$?)

1. Is there any place summarizing the known results about rings with the 'flat elements'? (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?)

(the world 'flat' is misleading, maybe there is some other name for the elements of $\mathfrak{m}^\infty$ ?)