Difference between revisions of "Fractional ideal"
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Latest revision as of 20:05, 23 January 2017
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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