Fractional ideal
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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