Unqiue Factorization Domains, is the product finite?

Question

Having looked around a bit, the most common definition of a UFD is an integral domain such that any element can be expressed as a product of a unit and irreducible elements, and that this representation is unique up to order of the product and associates of the irreducible elements. When I'm working with this definition though, and I need to use this unique representation, am I not being general enough if I assume this sum is finite? I know in the case of the integers, the product is always finite, as any infinite product of elements not equal to 1 is not going to be finite. Is there any way to extend that argument to a general UFD?

Thanks for any help.

Answer 1

(To make sure this question doesn't hang around unanswered...)

In every definition of UFD that I've seen, the fact the product is finite is always explicitly mentioned (although it is implicit because infinite products are not defined in general rings.)

To state uniqueness of a decomposition, an author is usually obliged to write this:

If $a=p_1p_2\ldots p_n=q_1q_2\ldots q_m$ are two factorizations of $a$ where $p_i, q_i$ are irreducibles and $n,m\in \Bbb Z^+$, then $n=m$ and after permutation the $p_i$'s and $q_i$'s pair up as associates.