Say we have a UFD A and a prime ideal P. Is A/P a UFD?
2026-03-27 20:21:07.1774642867
Is the property of being a UFD preserved under quotients by prime ideals?
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Not necessarily. $\mathbb C[X,Y,Z]$ is a UFD, but $$\mathbb C[X,Y,Z]/(Z^2-XY)=\mathbb C[x,y,z] $$ is not.
Edit
Although it is certainly possible to give an elementary proof of non UFDness based on the two different factorizations $z^2=xy$ of the same element, I can't resist the temptation to say that $\mathbb C[x,y,z]$ is not a UFD because its class group is $\mathbb Z/(2)$ (Hartshorne, example 6.5.2, page 133), whereas a UFD has zero class group (Hartshorne, Proposition 6.2, page 131).