I was given an exercise to show $A=\frac{\mathbb{C}[X,Y]}{(X^2+Y^2-1)}$ is a PID. But I wonder if it is true. Note that PID $\implies$ UFD. But we have $$X\cdot X = 1-Y^2 =(1-Y)(1+Y)$$ in $A$ which contradicts UFD.
Is there something wrong in the above factorization? Any suggestions / hints.
Edit: I can prove mechanically it is PID. My main concern was the above factorization. Thanks for the comments.