Let (R,m) be a local ring over a field. Suppose the ring has flat elements, i.e. m∞≠{0}. (The prototype is of course C∞(Rp,0), or a quotient of it, by some finitely generated ideal.)

~~1. For which rings and ideals, J⊂R, the following holds. If the completions satisfy ˆJ⊇(ˆm)N then J⊇mN+n, for some finite n. At least J⊇m∞?~~(Probably a better/more important version: suppose the image of J, under the completion map R→ˆR, contains (ˆm)N. Do we have J⊇mN? This seems to hold when mN is finitely generated)

- Is there a notion of the ‘infinite-radical’ of an ideal, ∞√m∞ ? I would like in the ‘geometric’ case ∞√m∞ to be the defining ideal of the set V(m∞). For example, for the ring R=k[[y_]]⊗C∞(Rp,0) we have m∞=(x_)∞, here x_ are the local coordinates on (Rp,0). Thus ∞√m∞=(x_).
(upd: What is the definition of the height of m∞?)

- Is there any place summarizing the known results about rings with the ‘flat elements’? (Maybe extending some properties of flat elements of C∞(Rp,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 m∞ ?)

**Answer**

**Attribution***Source : Link , Question Author : Dmitry Kerner , Answer Author : Community*