Let K be a number field, let R be a number ring in K, and let I R be an ideal.

a. Show that the set r (I) = {x K | xI I} is a subring of K.

b. Show that if I is invertible (Hint), then r (I) = R.

c. Show that for every subgroup A of (K, +) there is at most one order R in K, with respect to which A is an invertible ideal.

Hint: Let I be a fractional ideal of a domain R. Then I is called invertible if II^-1 = I^-1I = R. By the above, this is the case if and only if I J = JI = R for some fractional ideal J of R. Right now, we only know that II^-1 = I^-1I = R for principal ideals I. So far we therefore only know these ideals to be invertible.