A Dedekind domain is a CDR

A Dedekind domain $𝑅$ is also a CDR, #m/thm/ring i.e. given ideals $𝔞, 𝔟 ⊴ 𝑅$ we have¹

𝔞 ∣ 𝔟 ⟺ 𝔟 \subseteq 𝔞

Proof

The forward direction already holds in general (vide ^D1).
Since Fractional ideals of a Dedekind domain form an abelian group, if $𝔟 \subseteq 𝔞$ then $𝔠 : = 𝔟 𝔞 - 1 \subseteq 𝔞 𝔞 - 1 = 𝑅$ so $𝔠 ⊴ 𝑅$ and $𝔞 𝔠 = 𝔟$ .

#state/tidy | #lang/en | #SemBr

2022. Algebraic number theory course notes, ¶1.39, p. 20 ↩

A Dedekind domain is a CDR

Footnotes