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Let $R$ be a Noetherian domain; let ${I_{(i)}}$ be a set of fractional ideals in $K$, the fraction field of $R$, indexed by a lattice such that $I_{(i)} I_{(j)} = I_{(i + j)}$. Let $J_{(i)} = I_{(i)} \cap R$.

Is it possible to recover $I_{(i)}$ from knowing all the $J_{(i)}$?

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  • $\begingroup$ What are the restrictions on your indexing set? What do you mean by a "lattice" in this context? $\endgroup$
    – Neil Epstein
    Commented Aug 3, 2014 at 10:35
  • $\begingroup$ Lattice here would just mean a finite dimensional free abelian group $\mathbb{Z}^n$. It does have to be the full lattice (and not a subcone), as otherwise you could pick fractional ideals including $R$. $\endgroup$
    – user44191
    Commented Aug 3, 2014 at 18:24

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