For a domain $R$ to be a Dedekind domain it need to satisfy 3 conditions: one-dimensional, Noetherian, integrally closed.

I have got three domains satisfying all but one of those three:

1) $\mathbb{C}[x,y]$: not one-dimensional

2) Ring of all algebraic integers: not Noetherian

3) $\mathbb{Z}[\sqrt{-3}]$: not integrally closed

I know an equivalent definition of Dedekind domain is that all ideals of it can be factored into product of prime ideals.

Can anyone show me an ideal in 2) and 3) that cannot be factored into primes?