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For questions about factorization, the decomposition of mathematical objects (e.g. natural numbers, polynomials) into products of smaller objects (e.g. primes, lower degree polynomials).
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Seeking Noetherian normal domain with vanishing Picard group but not a UFD
Here is a very simple way to find an example: If $R$ is local, then the Picard group is always trivial (because locally free = free here).
So basically you want a local normal domain which is not UF …