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
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