Can you give or construct an elementary example of a factorial ring with elements which are product of infinitely many irreducible elements? i.e. there are reducible elements that can't be written as a finite product of irreducible ones
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1$\begingroup$ I think the ring of entire functions on $\mathbb{C}$ should work thanks to the Weierstrass factorization theorem. For instance, $\displaystyle \sin(z) = z \prod_{\substack{n \in \mathbb{Z}\\ n \neq 0}} \left(1 - \frac{z}{n \pi}\right)$. $\endgroup$– Viktor VaughnCommented Oct 21, 2018 at 15:29
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$\begingroup$ @André3000 formal power series?Every element in that ring is reduced into a product of and an invertible element and/or an irreducible one, because any element with an invertible constant monomial is invertible. $\endgroup$– SnateCommented Oct 21, 2018 at 17:35
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1$\begingroup$ I thought factorial ring is a synonym for UFD, so that factorizations can on have finitely many terms. $\endgroup$– rschwiebCommented Oct 21, 2018 at 19:27
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$\begingroup$ @Snate No, not formal power series, entire complex functions, as I said. Only power series with an infinite radius of convergence define entire functions. For instance, if you try to invert $1 - \frac{z}{n \pi}$, I think you'll see that its radius of convergence is much smaller than $\infty$. $\endgroup$– Viktor VaughnCommented Oct 22, 2018 at 2:35
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$\begingroup$ @Snate By "Factorial ring" do you mean something other than a UFD? $\endgroup$– rschwiebCommented Oct 23, 2018 at 16:38
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an elementary example of a factorial ring with elements which are product of infinitely many irreducible elements?
Firstly, a "factorial ring" is a synonym for a unique factorization domain, in which factorizations are finite and required to be unique (up to units.)
Secondly, infinite products are not defined for rings in general, so we do not usually talk about them. Perhaps you can share a certain context with us that you are interested in where infinite products are defined. I'm pretty sure they exist, but I'm not going to pick a random one.
Finally, you can certainly ask about cases where factorizations are not unique: that is, they may have factorizations with distinct irreducibles, or they may even have different numbers of irreducibles in the factorization.
A domain in which every nonunit, nonzero element has a (finite) factorization into irreducibles is called an atomic domian. At the same link you can read BFD's, HFD's and FFD's, all of which explore the different possibilities for decomoposing unique factorization into smaller concepts.
answered Oct 23, 2018 at 17:59
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1$\begingroup$ Here are some atomic domains that aren't UFD's at DaRT $\endgroup$– rschwiebCommented Oct 23, 2018 at 18:00