In the definition of an UFD $R$, it is said that $R$ should in particular be integral. For me, this hints that people are not so interested in rings that are UFD except for the fact that they are not integral domains. Why is that?
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No, I don't think this has to do with what people are interested. It is just a definition, requiring more than only integral domain. This is important for rings of integers of number fields, which are PIDs iff they are UFDs. – Dietrich Burde Nov 11 '18 at 19:39
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Try looking here. https://math.stackexchange.com/q/1797291/90543 – jgon Nov 11 '18 at 19:51
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For generalizations of the notion of UFD to rings with zero-divisors see the links I gave here.. There is not a unique generalization in use. – Bill Dubuque Nov 11 '18 at 20:25