Is the ideal $I=(X+Y,X-Y)$ in the polynomial ring $\mathbb{Z}[X,Y]$ a prime ideal ?Justify your answer.
We know I is a prime ideal if and only if Z/I is an integral domain now an integral domain has no divisor of zero But how to proceed after this please help!
Note that $\Bbb{Z}[X,Y]/(X-Y)\cong\Bbb{Z}[T]$ because the ring homomorphism $$\Bbb{Z}[X,Y]\ \longrightarrow\ \Bbb{Z}[T]:\ P(X,Y)\ \longmapsto P(T,T),$$ is surjective and has kernel $(X-Y)$. Therefore $$\Bbb{Z}[X,Y]/I\cong(\Bbb{Z}[X,Y]/(X-Y))/(\overline{X}+\overline{Y})\cong\Bbb{Z}[T]/(2T),$$ which is not an integral domain because $2T=0$ whereas $2,T\neq0$. So $I$ is not prime.