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Let $R$ be some integral domain and consider the following property: For all $a,x,y\in R$, if $a\mid xy$, then there exist $b\mid x$ and $c\mid y$ with $a=bc$.

Question: Is there a 'simple' classification of integral domains with this property?

Own work: Note that the property implies that irreducible elements are prime, so if $R$ satisfies the ascending chain condition on principal ideals, it must be a UFD. Since the property clearly holds in any UFD, the only 'interesting' $R$ are those that satisfy the property, but not the ascending chain condition on principal ideals.

Perhaps one can prove that the property is equivalent with irreducible elements are prime, in which case it holds precisely when $R$ is a GCD domain. This would be the type of 'simple' classification I'm looking for.

Mastrem
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    Isn't this what is called a “pre-Schreier domain” (https://en.wikipedia.org/wiki/Schreier_domain)? Not that I know much about it, but it might give some ideas for reading. – Hans Lundmark Sep 13 '19 at 14:55
  • @HansLundmark Good find! On the one hand, it gives me a good search term and indeed pre-Schreier domains are exactly the domains I'm looking for, so if you post it as an answer, maybe I should accept it. On the other hand, the classification is trivial since this is precisely the definition of a pre-Schreier domain and I'd be more interested in a connection to more 'well-known' classes of rings, such as GCD domains. – Mastrem Sep 13 '19 at 15:05
  • There is an example of a pre-schreier domain that isn’t a GCD domain here http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.506.4504&rep=rep1&type=pdf – rschwieb Sep 13 '19 at 15:34
  • @BillDubuque Thank you for linking the duplicate. The summary of domain properties there is very useful. – Mastrem Sep 13 '19 at 16:00
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    @Mastrem Glad it was helpful. I just found and added another dupe, which may also be of interest. Now that you know the terminology you should be able to locate much related literature, – Bill Dubuque Sep 13 '19 at 16:06

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