How to prove that $\Bbb C[X,Y]/(XY) \ncong \Bbb C[X,Y]/(X) \times \Bbb C[X,Y]/(Y)$

How to prove that $\Bbb C[X,Y]/(XY) \ncong \Bbb C[X,Y]/(X) \times \Bbb C[X,Y]/(Y)$ .

I have no idea about this problem on how to proceed, so I couldn't make any attempt.

• Does * denote the Cartesian product? – alphacapture Aug 12 '18 at 12:36
• Yes * denotes cartesian product – reflexive Aug 12 '18 at 12:41
• Are you confused about what the question is asking or is it just that you can’t figure out how to start? – alphacapture Aug 12 '18 at 12:55
• I understand the Question but can't solve it – reflexive Aug 12 '18 at 13:05
• @Mustafa, $\mathbb C[X] \times \mathbb C[Y]$ is not an integral domain. – lhf Aug 12 '18 at 21:25

If $\mathbb C[X,Y]/(XY)$ had a nontrivial idempotent, then there would exist polynomials $P(X)$, $Q(Y)$ with $$XY \mid (P(X)+Q(Y))^2 - (P(X)+Q(Y))$$ and $P(X)+Q(Y) \neq 0,1$. Looking at the highest degree coefficients, we get a contradiction.