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One can prove that if $A$ is a noetherian ring and an integral domain then we can write every $a\not\in A ^\times$, $a\neq0$, as a product of finite irreducible elements.

So I am looking for an example of a ring $A$, such there exist at least one irreducible element and there exist $a\not\in A^\times$ that we can't decomposed as a finite product of irreducible elements.

I tried to look at $\mathcal{C}^0(\mathbb{R},\mathbb{R})$ but it doesn't seen to work ($f$ inversible $\iff$ $f$ does not vanish).

Any help will be greatly appreciate

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The ring of holomorphic functions on $\mathbb C$ is a domain with elements which are divisible by infinitely many different primes.

There are indeed irreducible elements, namely those functions of the form $x-\alpha$ for $\alpha\in \mathbb C$.

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