Rings where ideals decompose into product of primes

Question

In a Dedekind Domain we have unique decomposition of ideals into product of prime ideals. My question out of curiosity is: Is there a ring satisfying this which is not a Dedekind Domain?

Answer 1

Something that looks like a Dedekind Domain but is not a domain.

In particular every $\mathbb{Z}/n\mathbb{Z}$ with $n$ square-free should work, by the Chinese remainder theorem.

Answer 2

Any integral domain with unique ideal factorization is already a Dedekind domain, see here. For rings with zero divisors, this is not true in general - consider $\mathbb{Z}/6$ for example.