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Concept
Fractional ideal
Summary
In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral domain are like ideals where denominators are allowed. In contexts where fractional ideals and ordinary ring ideals are both under discussion, the latter are sometimes termed integral ideals for clarity.
Let be an integral domain, and let be its field of fractions.
A fractional ideal of is an -submodule of such that there exists a non-zero such that . The element can be thought of as clearing out the denominators in , hence the name fractional ideal.
The principal fractional ideals are those -submodules of generated by a single nonzero element of . A fractional ideal is contained in if, and only if, it is an ('integral') ideal of .
A fractional ideal is called invertible if there is another fractional ideal such that
where
is called the product of the two fractional ideals).
In this case, the fractional ideal is uniquely determined and equal to the generalized ideal quotient
The set of invertible fractional ideals form an abelian group with respect to the above product, where the identity is the unit ideal itself. This group is called the group of fractional ideals of . The principal fractional ideals form a subgroup. A (nonzero) fractional ideal is invertible if, and only if, it is projective as an -module. Geometrically, this means an invertible fractional ideal can be interpreted as rank 1 vector bundle over the affine scheme .
Every finitely generated R-submodule of K is a fractional ideal and if is noetherian these are all the fractional ideals of .
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 set of fractional ideals over a Dedekind domain is denoted .
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