Redundancy in the definition of Dedekind domain?

Question

Is there a domain which is noetherian and whose nonzero prime ideals are maximal, but which is not integrally closed?

This may be a silly question to experts. I ask because I think I have found proofs for the following implications:

1. Noetherian domain whose nonzero prime ideals are maximal $\implies$

2. Every fractional ideal is projective/invertible $\implies$

3. Fractional ideals form an abelian group under multiplication $\implies$

4. Integrally closed noetherian domain whose nonzero prime ideals are maximal

But I can't find an error on first inspection.

Answer 1

Yes, there are such domains: for instance, $\mathbb Z[\sqrt{-3}]$ or $k[X^2,X^3]$. These are not integrally closed, but they are noetherian of dimension one.