Ring theory MOC

Fractional ideal

A fractional ideal is a generalization of an ideal in the same way Rational numbers generalizes Integers. Let 𝑅 be an Integral domain, and 𝐾 =Frac⁡𝑅 its field of fractions. A fractional ideal 𝔵 ≤𝑅𝐾 is an 𝑅-submodule of 𝐾 such that 𝑟𝔵 ⊆𝑅 for some 𝑟 ∈𝑅. 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,𝔟 ≤𝑅𝐾

(𝔞𝔟)=(𝔞:𝐾𝔟)={𝑥∈𝐾:𝔟𝑥⊆𝔞}

A fractional ideal 𝔞 is invertible iff (𝔞𝔞−1) =(1) for some (provably unique) inverse fractional ideal 𝔞−1 ≤𝑅𝐾, which if it exists is given by

𝔞−1=((1)𝔞)

Proof

For uniqueness of the inverse, note if 𝔞𝔟 =(1) =𝔞𝔟′ then 𝔟 =𝔟(1) =𝔟𝔞𝔟′ =(1)𝔟′ =𝔟′. Suppose now 𝔞 is invertible Then 𝑏𝔞 ⊆(1) for all 𝑏 ∈𝔞−1, so 𝔞−1 ⊆((1)𝔟). Thus

(1)=𝔞𝔞−1⊆𝔞((1)𝔞)⊆(1)

whence

𝔞((1)𝔞)=(1)

as required. Clearly the inverse is a fractional ideal, as for any 0 ≠𝑎 ∈𝐼 we have 𝑎𝐼−1 ⊆𝑅.

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