Given an integral domain with field of fractions , a fractional ideal, , of is an -submodule of such that for some nonzero . Explicitly, is a subset of such that for any and :
To prevent confusion, regular ideals of are sometimes called integral ideals. Clearly any integral ideal of is a fractional ideal of (simply take ). Moreover, it is easy to see that a fractional ideal, , of is an integral ideal of iff .
Addition and multiplication of fractional ideals can now be defined just as they are for integral ideals. Namely, for fractional ideals and we let be the submodule of generated by the set and let be the submodule of generated by the set . Note that if then , so these are indeed fractional ideas.
Fractional ideals are of great importance in algebraic number theory, specifically in the study of Dedekind domains.
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