rings with ‘flat functions’

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, JR, the following holds. If the completions satisfy ˆJ(ˆm)N then JmN+n, for some finite n. At least Jm? (Probably a better/more important version: suppose the image of J, under the completion map RˆR, contains (ˆm)N. Do we have JmN? This seems to hold when mN is finitely generated)

  1. 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?)

  1. 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 ?)


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

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