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?




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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.



Factorization of elements vs. of ideals, and is being a UFD equivalent to any property which can be stated entirely without reference to ring elements?
This is sort of an anti-answer, but: my instinct is that ZC is taking the categorical perspective too far. 


To start philosophically, I think it is quite appropriate to, when given a mathematical structure like a topological space or a ring -- i.e., a set with additional structure -- refrain from inquiring as to exactly what sort of object any element of the structure is. There is a famous essay "What numbers could not be" by Paul Benacerraf, in which he pokes fun at this idea by imagining two children who have been taught about the natural numbers by two different "militant logicists". Their education proceeds well until one day they get into an argument as to whether 3 is an element of 17. (The writing is very nice here and unusually witty for an essay on mathematical philosophy: the names of the children are Ernie and Johnny, an allusion to Zermelo and von Neumann, who had rival definitions of ordinal numbers.) The point of course is that it's a silly question, and a mathematically useless one: it won't help you to understand the structure of the natural numbers any better.


On the other hand, to deny that a set is an essential part of certain (indeed, many) mathematical structures seems to be carrying things too far. As far as I know, it is not one of the goals of category theory to eliminate sets (though one occasionally hears vague mutterings in this direction, I have never seen an explanation of this or, more critically, of the need for this). 


Coming back to rings, it seems to me that very few properties of rings can be expressed without elements. You also seem to implicitly suggest that it is "more structural" to think about things in terms of ideals than elements. Can you explain this? It would seem that speaking of ideals involves *more* set theoretic machinery than speaking about elements: this is certainly true in model theory in the language of rings. 


It seems wrong to say that unique factorization of ideals into primes is a "generalization" of unique factorization of elements, since neither property implies the other. 


Finally a positive remark: it sounds like you might like the characterization of UFDs as Krull domains with trivial divisor class group. 


