8410Krumm--Dedekinddomains

8410Krumm--Dedekinddomains - Dedekind Domains with Torsion...

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Dedekind Domains with Torsion Class Group David Krumm Let R be a Dedekind domain with fraction field K . An overring of R is any ring A such that R A K . I will prove the following result: The class group of R is a torsion group if and only if every overring of R is a localization of R . Suppose first that the class group of R is torsion, and let A be any overring of R . We know that A = \ p T R p ( ? ) for some set T of prime ideals of R . Explicitly, T = { p : p A 6 = A } . I will show that A = S - 1 R where S is the multiplicative set R \ ∪ p T p . It is clear from the definition of localization that S - 1 R A . To see the reverse inclusion, fix an element a A and consider the ideal D = { r R : ra R } of denominators of a . Since the class group of R is torsion, there is a positive integer n such that D n is a principal ideal, say D n = ( s ). Thus for any prime ideal p of R we have that p divides D if and only if s p . Since D intersects R \ p for all
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