Ring with infinitely reducible elements 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
 A: 
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.
