Ring theory MOC

Fractional ideal

A fractional ideal is a generalization of an ideal in the same way Rational numbers generalizes Integers. Let be an Integral domain, and its field of fractions. A fractional ideal is an -submodule of such that for some . ring Thus fractional ideals are proper ideals divided by a nonzero elements.

Ideal quotient

The ideal quotient, which in these notes refers to the generalized colon ideal, is defined as follows for fractional ideals

A fractional ideal is invertible iff for some (provably unique) inverse fractional ideal , which if it exists is given by

Proof

For uniqueness of the inverse, note if then . Suppose now is invertible Then for all , so . Thus

whence

as required. Clearly the inverse is a fractional ideal, as for any we have .

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