Characterisation of UFDs

Theorem
Let $A$ be an integral domain.

Then the following are equivalent:


 * $(1): \quad A$ is a unique factorisation domain


 * $(2): \quad A$ is a GCD domain satisfying the ascending chain condition on principal ideals.


 * $(3): \quad A$ satisfies the ascending chain condition on principal ideals and every irreducible element of $A$ is a prime element of $A$.