Dedekind Domains
In Dedekind domains, the situation is much simpler. In particular, every non-zero fractional ideal is invertible. In fact, this property characterizes Dedekind domains: an integral domain is a Dedekind domain if, and only if, every non-zero fractional ideal is invertible.
The quotient group of fractional ideals by the subgroup of principal fractional ideals is an important invariant of a Dedekind domain called the ideal class group. Part of the reason for introducing fractional ideals is to realize the ideal class group as an actual quotient group, rather than with the ad hoc multiplication of equivalence classes of ideals.
Read more about this topic: Fractional Ideal
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