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.
Definition and basic results
Let R be an integral domain, and let K be its field of fractions. A fractional ideal of R is an R-submodule I of K such that there exists a non-zero r ∈ R such that rI ⊆ R. The element r can be thought of as clearing out the denominators in I. The principal fractional ideals are those R-submodules of K generated by a single nonzero element ofK. A fractional ideal I is contained in Rif, and only if, it is an ideal of R. A fractional ideal I is called invertible if there is another fractional ideal J such that IJ = R. In this case, the fractional ideal J is uniquely determined and equal to the generalized ideal quotient The set of invertible fractional ideals form an abelian groupwith respect to the above product, where the identity is the unit idealR itself. This group is called the group of fractional ideals of R. The principal fractional ideals form a subgroup. A fractional ideal is invertible if, and only if, it is projective as an R-module. Every finitely generatedR-submodule of K is a fractional ideal and if R is noetherian these are all the fractional ideals of R.
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 set of fractional ideals over a Dedekind domain is denoted. Its 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.
Number fields
Recall that the ring of integers of a number field is a Dedekind domain. We call a fractional ideal which is a subset of integral. One of the important structure theorems for fractional ideals of a number field states that every fractional ideal decomposes uniquely up to ordering as for prime ideals. For example, factors as Also, because fractional ideals over a number field are all finitely generated we can clear denominators by multiplying by some to get an ideal. Hence Another useful structure theorem is that integral fractional ideals are generated by up to elements. There is an exact sequence associated to every number field, where is the ideal class group of.
Examples
is a fractional ideal over
In we have the factorization. This is because if we multiply it out, we get
Since satisfies, our factorization makes sense.
In we can multiply the fractional ideals and to get the ideal
Divisorial ideal
Let denote the intersection of all principal fractional ideals containing a nonzero fractional ideal I. Equivalently, where as above If then I is called divisorial. In other words, a divisorial ideal is a nonzero intersection of some nonempty set of fractional principal ideals. If I is divisorial and J is a nonzero fractional ideal, then is divisorial. Let R be a local Krull domain. Then R is a discrete valuation ringif and only if the maximal ideal of R is divisorial. An integral domain that satisfies the ascending chain conditions on divisorial ideals is called a Mori domain.
