Dedekind domain

Prime ideals are invertible in a Dedekind domain

Let be an ideal, be a nonzero prime ideal, and be a nonzero ideal. Then .¹ ring In particular, .

Proof

Let be the field of fractions.

First we consider the case , i.e. we must show , whereby it is sufficient to find . By definition, iff .

We will try to find so that , but , so that is the appropriate . To this end, let . By ^P1, we have for some nonzero prime ideals, where we are free to assume that is minimal. Since it follows by ^D2 . say . But since , is maximal, so .

If , then again by maximality , whence cannot be equal to or else is a unit and which is not prime.

Now consider , whence by minimality of . Hence there exists such that . By construction , and it follows that , proving the case .

More generally, use the Noetherian nature of to write . Suppose towards contradiction . Then for every , we may write

for some . Let , so that

whence . But is a monic polynomial in with coëfficients in , whence is integral over . But is integrally closed, hence , contradicting the above special case.

For invertibility, note for all , and , so . Since but is an ideal, and is maximal, it follows .

This is really just a lemma for the further-reaching fact Fractional ideals of a Dedekind domain form an abelian group.

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Footnotes

1. 2022. Algebraic number theory course notes, ¶1.35–1.36, pp. 18–19 ↩