# Characterisation of UFDs

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## 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 $A$.