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For fractional ideals of a Dedekind Domain, are each of the elements that generate the ideal (ie. form the basis of the lattice associated with the ideal) always scaled by the same amount? That is to say, scaled by the same element from the field of fractions?

Mike Four
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I'm not sure if I understand the question, but a fractional ideal is a fraction times an actual ideal, by the definition, i.e. for a fractional ideal $I$ over a ring $R$ we have some $r\in R$ such that $rI\unlhd R$.

tomasz
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  • For instance. What about the ideal (2,1+sqrt(-5)) in Z[sqrt(-5)]? If you multiply the ideal by the fraction 1/(1+sqrt(-5)) you get the ideal (1/3(1-sqrt(-5)),1). Would you then get all the elements of the ring R=Z[sqrt(-5)] as well as a principal fractional ideal of (1/3(1-sqrt(-5))? Or would you just end up with 1/3 times R? – Mike Four Feb 08 '14 at 03:12