# Characterisation of UFDs

## Theorem

Let $A$ be an integral domain.

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 $A$.