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 $A$.


Proof