Characterisation of UFDs

Theorem
Let $A$ be an integral domain.

Then the following are equivalent:


 * 1. $A$ is a unique factorisation domain


 * 2. $A$ is a GCD domain satisfying the ascending chain condition on principal ideals.


 * 3. $A$ satisfies the ascending chain condition on principal ideals and every irreducible element of $A$ is prime.