Ring homomorphisms Algebraic structures Related structures Algebraic number theory Noncommutative algebraic geometry Free algebra Clifford algebra 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.
generated by a single nonzero element of
is called invertible if there is another 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
The principal fractional ideals form a subgroup.
A (nonzero) fractional ideal is invertible if and only if it is projective as an
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
In Dedekind domains, the situation is much simpler.
In particular, every non-zero fractional ideal is invertible.
In fact, this property characterizes Dedekind domains: The set of fractional ideals over a Dedekind domain
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.
For the special case of number fields
Hence the theory of fractional ideals can be described for the rings of integers of number fields.
In fact, class field theory is the study of such groups of class rings.
and the subgroup of principal fractional ideals is denoted
The ideal class group is the group of fractional ideals modulo the principal fractional ideals, so and its class number
In some ways, the class number is a measure for how "far" the ring of integers
is from being a unique factorization domain (UFD).
There is an exact sequence associated to every number field.
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 in the spectrum of
For example, Also, because fractional ideals over a number field are all finitely generated we can clear denominators by multiplying by some
Hence Another useful structure theorem is that integral fractional ideals are generated by up to 2 elements.
We call a fractional ideal which is a subset of
denote the intersection of all principal fractional ideals containing a nonzero fractional ideal
[2] In other words, a divisorial ideal is a nonzero intersection of some nonempty set of fractional principal ideals.
Let R be a local Krull domain (e.g., a Noetherian integrally closed local domain).
Then R is a discrete valuation ring if and only if the maximal ideal of R is divisorial.
[3] An integral domain that satisfies the ascending chain conditions on divisorial ideals is called a Mori domain.