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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.

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

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