product of two elements that factor in a unique way factors in a unique way?

Question

I asked myself a question that I can't really answer. If I have a ring R and two elements $x$ and y that factor in a unique way into irreductible elements $x=x_1 \ldots x_m$ and $y=y_1\ldots y_m$ does this imply that $xy$ factors uniquely as $xy=x_1\ldots x_n y_1 \ldots y_m$ or does the fact that $x$ and $y$ factor in a unique way is not sufficient (i.e. we need R to be a UFD)?

Answer 1

This is not enough. Indeed, there are examples when $x$ and $y$ are irreducible (then the decomposition is obviously unique), and there is another decomposition $xy = pq$. Take for example $x=2$ and $y=3$ in $\mathbb Z[i \sqrt{5}]$, and the well-known example: $$2 \cdot 3 = (1+i \sqrt{5}) \cdot (1-i \sqrt{5}).$$

Answer 2

In $\mathbb{Z}[\sqrt5\,i]$, all elements $2$, $3$, $1+\sqrt5\,i$ and $1-\sqrt5\,i$ are irreducible. However,$$2\times3=\left(1+\sqrt5\,i\right)\times\left(1-\sqrt5\,i\right).$$