number theory - Rings where divisors of $mn$ are product of divisors of $m$ and $n$; relation to UFDs

Using the fundamental theorem of arithmetic, it's easy to prove this proposition:

Proposition. Every divisor of $mn$ can be written as the product of a divisor of $m$ to a divisor of $n$.

My question: How heavily does the proposition rely on the fundamental theorem of arithmetic? Is there anyway to prove it with this theorem or one of its equivalents? What happens in rings which prime factorization is not unique?