Fractional ideal

A fractional ideal is a generalization of an ideal in the same way $ℚ$ generalizes $ℤ$ . Let $𝑅$ be an Integral domain, and $𝐾 = F r a c 𝑅$ its field of fractions. A fractional ideal $𝔵 \leq 𝑅 𝐾$ is an $𝑅$ -submodule of $𝐾$ such that $𝑟 𝔵 \subseteq 𝑅$ for some $𝑟 \in 𝑅$ . #m/def/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 $𝔞, 𝔟 \leq 𝑅 𝐾$

(𝔞 𝔟) = (𝔞 : 𝐾 𝔟) = {𝑥 \in 𝐾 : 𝔟 𝑥 \subseteq 𝔞}

A fractional ideal $𝔞$ is invertible iff $(𝔞 𝔞 - 1) = (1)$ for some (provably unique) inverse fractional ideal $𝔞 - 1 \leq 𝑅 𝐾$ , which if it exists is given by

𝔞 - 1 = ((1) 𝔞)

Proof

For uniqueness of the inverse, note if $𝔞 𝔟 = (1) = 𝔞 𝔟'$ then $𝔟 = 𝔟 (1) = 𝔟 𝔞 𝔟' = (1) 𝔟' = 𝔟'$ . Suppose now $𝔞$ is invertible Then $𝑏 𝔞 \subseteq (1)$ for all $𝑏 \in 𝔞 - 1$ , so $𝔞 - 1 \subseteq ((1) 𝔟)$ . Thus

(1) = 𝔞 𝔞 - 1 \subseteq 𝔞 ((1) 𝔞) \subseteq (1)

whence

𝔞 ((1) 𝔞) = (1)

as required. Clearly the inverse is a fractional ideal, as for any $0 \neq 𝑎 \in 𝐼$ we have $𝑎 𝐼 - 1 \subseteq 𝑅$ .

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