Proof of that entire $A$ is a UFD iff every nonzero prime ideal contains a nonzero prime element. (Kaplansky)

Question

This is called Kaplansky condition: For $A$ entire, $A$ is a UFD iff every nonzero prime ideal contains a nonzero prime element.

Here prime element means that $p \in A$ is prime iff $p$ is non-zero non-unit and $p\mid ab$ implies $p \mid a$ or $p\mid b$.

Where can I get the proof of this statement?

Answer 1

Kaplansky's book "Commutative Rings" has the appropriately correct statement in the first few pages.