Fractional ideal

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In mathematics, in particular commutative algebra, the idea of fractional ideal is introduced in the context of Dedekind domains. In general commutative rings there is no guarantee that one can divide an ideal I by another one J that is non-zero, and get a satisfactory inverse to multiplication of ideals. But that is possible under special circumstances, that play an important part in algebraic number theory. This construction is also used in the theory of algebraic curves. In contexts where fractional ideals and ordinary ring ideals are both under discussion, the latter are sometimes termed integral ideals for clarity.

The definition of fractional ideal relies on the definitions of module and field of quotients. Let R be a Dedekind domain. It is an integral domain, and so it possesses a field of quotients K. A fractional ideal of R is a nonzero finitely generated R-submodule of K. A fractional ideal I is contained in R if and only if it is an ('integral') ideal of R.

Fractional ideals can be multiplied in a natural manner, and it can be shown that in a Dedekind domain, the fractional ideals form an abelian multiplicative group with R as identity element. The principal fractional ideals are those R-submodules of K generated by a single nonzero element of K. It is easy to see that they form a subgroup of the group of all fractional ideals.

The quotient group of fractional ideals divided by principal fractional ideals is isomorphic to the ideal class group of R. Part of the reason for introducing fractional ideals is to realize the ideal class group as an actual quotient group, rather than with the ad hoc multiplication of equivalence classes of ideals.

R-module theory

An alternative characterization of Dedekind domain is that every fractional ideal of R be invertible under ideal multiplication. From a more contemporary perspective, the fractional ideals of K are all projective modules over R, of rank 1. The multiplication operation on (isomorphism equivalence classes of) them is the tensor product of R-modules.

It is in fact relatively unusual for tensor product with a module to be an invertible operation; here the inverse in the class group taking I to (the class of) I-1 corresponds also to taking a module P to HomR(P,R). Each isomorphism class of rank-one projective R-module is in fact represented by some fractional ideal - or, clearing 'denominators' with a principal ideal, by an actual ideal of R. de:gebrochenes Ideal