Let $I=(X^2+Y^2)\subset\mathbb{R}[X,Y], A=\mathbb{R}[X,Y]/I$. And $x,y$ denotes $X+I,Y+I \in A/I$ respectively.
I proved that $I$ is a prime ideal of $\mathbb{R}[X,Y]$. So, there exists the quotient field of $A$ (denoted by $K$).
I need to compute the integral closure of $A$ in $K$ (denoted by $B$). I know that $y/x,x/y\in B$ because $(y/x)^2+1=(x/y)^2+1=0$. Since $x/y=-y/x$, we get $A[y/x,x/y]=A[y/x]\subseteq B$.
I believe that $B=A[y/x]$ but cannot prove the other inclusion. Tell me whether my expectation is true or false, and how to compute the integral closure?