Why exactly is the unique factorization of elements into irreducibles a natural thing to look for? Of course, it's true in $\mathbb{Z}$ and we'd like to see where else it is true; also, regardless of whether something is natural or not, studying it extends our knowledge of mathematics, which is always good. But the unique factorization of elements - being specifically a question of **elements** - seems completely counter to the category theory philosophy of characterizing structure via the maps between objects rather than their elements. Indeed, I feel like unique factorization of ideals into prime ideals is less a generalization of unique factorization of elements into irreducibles than the latter is a messier, unnatural special case of the former, a "purer" question (ideals, being the kernels of maps between rings, I feel meet my criteria for being a category-theoretically acceptable thing to look at). Certainly, the common theme in algebra (and most of mathematics) is to look at the decomposition of **structures** into simpler **structures** - but quite rarely at actual elements.

Now, for nice cases like rings of integers in number fields, we can characterize being a UFD in terms of the class group and other nice structures and not have to mess around with ring elements, but looking at the Wikipedia page on UFDs and the alternative characterizations they list for general rings, they all appear to depend on ring elements in some way (the link to "divisor theory" is broken, and I don't know what that is, so if someone could explain it and/or point me to some resources for it, it'd be much appreciated).

Sorry about the rambling question, but I was wondering if anyone had any thoughts or comments? Is "being a UFD" equivalent to any property which can be stated entirely without reference to ring elements? Should we care whether it is or not?

EDIT: Here's a more straightforward way of saying what I was trying to get at: The structure theorem for f.g. modules over a PID, the Artin-Wedderburn theorem, the Jordan-Holder theorem - these are structural decompositions. Unique factorization of elements is not, **because elements are not a structure**. My feeling is that this makes it a fundamentally less natural question, and I ask whether being a UFD can be characterized in purely structural terms, which would redeem the concept somewhat, I think.

elementof the Hom-set - factors uniquely as a "product" (again, having to give the maps R's ring structure) of "irreducible" maps is still much more ungainly, in my opinion, than whether ideals factor as a product of prime ideals, which is in the proud tradition of other structural decomposition questions such as whether a ring is semi-simple. $\endgroup$ – Zev Chonoles Dec 5 '09 at 2:43