Let $k$ be a field with $k[x_1,\ldots ,x_n]=:k[X]$ and $k[y_1,\ldots ,y_n]=:k[Y]$. Suppose $I$ is an ideal in $k[X,Y]$ such that $I=I_1+I_2$ where $ I_1$ and $I_2$ are ideals in $k[X]$ and $k[Y]$ respectively. If $P$ is a minimal prime ideal containing $I$, is it true that $P$ can be written as $P=P_1+P_2$ where $P_1$ and $P_2$ are minimal prime ideals of $I_1$ and $I_2$ in $k[X]$ and $k[Y]$ respectively? If so how we can prove this?
Any reference or hint will be helpful.
Thank you.
I don’t think it is true. Let $k = Q$ and consider $k[x],k[y]$ with $I_1 = (x^2+1)$ and $I_2 = (y^2+1)$. The ring $k[x,y]/(x^2+1,y^2+1)$ is not a domain.