# Quotient ring is a UFD

Prove that the ring $\mathbb R[x,y,z]/(x^2+y^2+z^2-1)$ is a unique factorization domain.

• What do you expect the OP to say? "I checked $x^2$. I checked $x^2-1$. I checked $x^2-y^2$ ..." Or a confession that it is homework and they just don't want to do it? Just assume good faith. – OR. Sep 5 '13 at 18:55
• @ABC Brandon's request is standard practice when a question is posed like this. He is acting in good faith (but your response to him does not show very much good faith.) The OP could explain, for example, any line of approach they tried until this point, ideas, failures etc. – rschwieb Sep 5 '13 at 20:38
• You should get an idea from math.stackexchange.com/questions/244460 which discussed $\mathbb{R}[x,y]/(x^2+y^2-1)$. The same works here. – Martin Brandenburg Sep 5 '13 at 20:59
• Or you could try to show $\mathbb R /(x^2 +1)$ is not an UFD, which is isomorphic to $\mathbb C$. – Daron Sep 5 '13 at 21:07
• @MartinBrandenburg With two variables one can write $x\times x=(1-y)(1+y)$, providing two distinct irreducible factorizations. It is not clear at all whether a similar argument exists for three variables as you claim. Also several comments here conjecture that the ring is in fact a UFD – Ewan Delanoy Sep 6 '13 at 11:54

Hint. Let $R=\mathbb R[X,Y,Z]/(X^2+Y^2+Z^2-1)$. Note that $z-1$ is prime in $R$. Show that $R_{z-1}$ is a UFD and then use Nagata Criterion.
• What is $R_{z-1}$? – Sahiba Arora May 22 '17 at 20:04