Right now I’m teaching a class on unique factorization and its failure. One of my hobbies is finding fun problems like “Find a domain where every finitely generated ideal is principle, but not every ideal is finitely generated.” Today one of my students found an answer to the following fun question (which was cute enough that I thought I’d pass it on to all of you).
A weaker statement than unique factorization is just that the number of irreducible factors is independent of factorization. For example, in , even though 6 factors nonuniquely as both of these factorizations have 2 irreducible factors. How badly can this fail? It turns out the answer is “really badly.” Find a domain R where every nonzero element factors as a unit times a product irreducibles, but which has a fixed element that can be written as a product of more than N irreducibles for any arbitrarily large N.