Principal Ideal Domain and Factorization

Question

If $A$ is a local domain such that each non-trivial ideal factors uniquely into primes then does it follow that $A$ must be a principal ideal domain?

Answer 1

Yes. It's pretty well known that an integral domain is a Dedekind domain (a one dimensional, Noetherian, integrally closed integral domain) if and only if it is not a field and every nonzero ideal can be expressed as a product of prime ideals.

A Dedekind domain with only finitely many prime ideals is automatically a principal ideal domain.

Answer 2

Unique factorisation of non-zero ideals into a product of prime ideals means $A$ is a Dedekind domain. It is well known Dedekind domains have Krull dimension $1$ and their localisation at non-zero prime ideals are discrete valuation domains.

Hence a local domain with such factorisation is a D.V.R. Its maximal ideal is principal, generated by an element $\pi$, and its ideals are generated by the $\pi^n,\enspace n\in\mathbf N$.