Dedekind domain

A Dedekind domain is a CDR

A Dedekind domain 𝑅 is also a CDR, ring i.e. given ideals 𝔞,𝔟 ⊴𝑅 we have¹

𝔞∣𝔟⟺𝔟⊆𝔞

Proof

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

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tidy | en | SemBr

Footnotes

1. 2022. Algebraic number theory course notes, ¶1.39, p. 20 ↩