Question: Is an integral domain in which every proper ideal is a product of prime ideals necessarily a Dedekind domain? Please provide a reference.
The first two sentences of the Wikipedia page for Dedekind domains are as follows: 'In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every non-zero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily unique up to the order of the factors.'
This suggests that the definition of a Dedekind domain (or one possible definition of it) is an integral domain in which every non-zero proper ideal is a product of prime ideals, and that this property alone implies unique factorisation. However I can't find any outside source that confirms that this is true. The source that Wikipedia cites to support this claim is remark 3.25 of these notes, which, as far as I can tell, doesn't prove what Wikipedia claims it does, since the definition of a Dedekind domain in those notes is not "an integral domain in which every non-zero proper ideal is a product of prime ideals", but rather "a noetherian, integrally closed integral domain in which every prime ideal is maximal", and it is not clear (to me) that the latter statement is implied by the former.
Of course I find a reference right after asking the question. I swear I was looking for a while...
Anyway, it's true, and a proof is on page 765 of Dummit-Foote abstract algebra. If I were less busy I would change the reference on the wikipedia page to this, but alas.