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.
2026-03-25 04:42:32.1774413752
How to prove that $\Bbb C[X,Y]/(XY) \ncong \Bbb C[X,Y]/(X) \times \Bbb C[X,Y]/(Y)$
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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.