General result for rings in which irreducible is prime?

Question

There is a general result that characterizes the rings in which prime is equivalent to irreducible?

Answer 1

If $A$ is a domain, primes are always irreducible. If $A$ is a noetherian domain, then irreducibles are primes if and only if $A$ is an UFD.

In fact, if $A$ is an UFD, it is easy to check that an irreducible is prime. On the converse, if $A$ is a noetherian domain, we can always factorize an element $x$ in irreducibles (one starts dividing $x=x_1\cdot x_2$ etc. and the process ends because $A$ is noetherian). Then, if irreducibles are prime, one can check that the factorization is unique up to order and invertibles.

Answer 2

Wikipedia says

In an integral domain, every prime element is irreducible, but the converse is not true in general. The converse is true for unique factorization domains (or, more generally, GCD domains).