Does factorisation always exist? In the definition of UFD, there are two parts: existence and uniqueness. It is well-understood that the uniqueness property is useful e.g. $\mathbb{Z}[\sqrt{-5}]$. But isn't it true that every integral domain must satisfy the existence property?
Existence: Each element in $R$ which is neither zero nor unit can be written as product of irreducibles.
But this is always true, if an element can not be decomposed as product then it is irreducible.

For another example, consider the ring of algebraic integers. If you try to factor $2$ in this ring, for instance, you'll see that $2 = (\sqrt{2})^2 = (\sqrt[3]{2})^3 = \cdots$ since $\sqrt[n]{2}$ is integral for each $n$.
