$yB$ is not a prime ideal in $B.$

Question

Let $B = \mathbb R[x,y]$ where $x^2 + y^2 = 1$ which is called the coordinate ring of the unit circle.

I am trying to prove that $yB$ is not a prime ideal in $B.$

I have the following information about $B$:

1- $B$ is an integral domain.

And I know that in an integral domain a prime ideal is defined as an ideal $I \neq R$ such that if $ab \in I,$ then either $a \in I$ or $b \in I.$

Still I do not know how to use that definition to prove that $yB$ is not a prime ideal.

Any help will be appreciated!

Answer 1

Let $B= \frac{\mathbb{R}[X,Y]}{(X^2+Y^2-1)}$ and $yB=\frac{(Y,X^2+Y^2-1)}{(X^2+Y^2-1)}$ when $y$ is the image of $Y$ in $B$. Then $\frac{B}{yB}=\frac{\mathbb{R}[X,Y]}{(Y,X^2+Y^2-1)}=\frac{\frac{\mathbb{R}[X,Y]}{(Y)}}{\frac{(Y,X^2+Y^2-1)}{(Y)}}=\frac{\mathbb{R}[X]}{(X^2-1)}$ which is not a domain. Hence, it is not a prime ideal.