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.

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    $\begingroup$ This is not true, e.g. in $\mathscr{O}(\mathbb{C})$ the irreducible elements are entire functions with exactly one zero. An entire function with infinitely many zeros, like $\sin$, can therefore not be written as a (finite) product of irreducibles. $\endgroup$ – Daniel Fischer Jul 22 '18 at 15:26
  • $\begingroup$ $O(\mathbb{C})$ is the set of all entire functions? $\endgroup$ – Upc Jul 22 '18 at 15:31

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 $n$.


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