Let R be a Dedekind domain whose residue fields are finite, and let K be the field of fractions of R. When S is a (non-empty) subset of K we write Int(S) for the subring of K[X ] consisting of all polynomials f (X ) in K[X] such that f (S ) R. We show that there exist fractional ideals J 0 , J 1 , ..., J n and monic polynomials
where V n is the K-space of polynomials of degree at most n in K[X ]. This generalises classic results on Int(R).
1997 Academic Press
Let R be a Dedekind domain with finite residue fields and field of fractions K. Let E be a (non-empty) subset of K. Write Int R (E ) or just Int(E ) for the subring of polynomials f (X ) in K[X ] such that f (E ) R. We are interested in the structure of the A-module of Int R (E ) (see [2,6,7]).
For a subset E of K we define the R-closure cl R (E) or just cl(E ) of E to be the set of all elements a # K such that f (a) # R for every polynomial
We say that E is R-closed if E=cl R (E) and E is R-fractional if there exists d # R, d{0, such that dE R. It is easy to prove that cl(cl(E))= cl(E ) and so we may consider only the case when E is R-closed. In [5] McQuillan proved that for every not R-fractional subset E of K, Int(E )=R.