Re: Two questions on dedekind domain!

1. If every prime ideal in an integral domain R is
invertible, then R
is dedekind.

Invertible ideals are finitely generated, hence all primes
of R are finitely generated. By a theorem of Cohen R then
is noetherian.
Invertible ideals are locally principal, thus for every prime
p the maximal ideal pR_p is principal. By Krull's theorem
pR_p thus has height one. It follows that every prime is
maximal.
Moreover R_p is a factorial ring, hence integrally closed.
Since R is the intersection of all R_p we get that R is
integrally closed. Point 9 then yields the assertion.

2. If R is noetherian and every maximal ideal of R is
invertible, then
R is dedekind.

I guess that we should assume R to be a domain here?
Then the proof is almost equal to that of 1. - we know already
that R is noetherian.

H
.

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