I'm studying algebra, and I have the following problem:
$R$ is the ring of polynomial functions on the unit circle. Thus $R = \mathbb{R}[X, Y] / (X^2 + Y^2 - 1)$. Put $x := X + I$, $y := Y + I \in R$.
I want to proof that $(x-1)$ and $(y-1)$ are not prime ideals in $R$ and that $R$ is not a unique factorisation domain.
I have already shown that $x-1$ and $y-1$ are irreducible in $R$, this might be useful in the proof.
I was thinking about proving it with contradiction, since I know that $I$ is a prime ideal iff $R/I$ is a domain. However, nothing seems to get anywhere. Any hints?
Thanks in advance!