Dedekind domain

A Dedekind domain admits UFI

Let be an integral domain. Then admits unique factorization of ideals if¹ is a Dedekind domain ring

Dedekind implies UFI

Let be a nonzero proper ideal. First we show that if a prime factorization exists, it is necessarily unique. Suppose

whence , so without loss of generality by ^D2. Since , is maximal, whence . Multiplying both sides by as a Product ideal gives

since Prime ideals are invertible in a Dedekind domain, so we can induce that the factorization is unique.

To prove existence, we use the Noetherian property and Prime ideals are invertible in a Dedekind domain. Let be the set of all ideals of for which there exists no prime factorization, and assume towards , whence there exists a maximal element . Now must be contained in a maximal ideal (which is prime), and since we have

Since Prime ideals are invertible in a Dedekind domain guarantees , it follows from the maximality of in that has a prime factorization

whence

is a prime factorization of , i.e. , a contradiction.

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Footnotes

1. It seems to be possible to strengthen this to an iff. ↩