Let $\mathbb{K}$ be a field. What is the Krull dimension of $\frac{\mathbb{K}[X_1,X_2]}{(X_1X_2)}$?
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A prime ideal of this quotient corresponds to a prime ideal of $K[X_1,X_2]$ containing one of $X_1$ or $X_2$. So $(X_1)/(X_1X_2)$ and $(X_2)/(X_1X_2)$ are the minimal prime ideal of the quotient.
Furthermore, $$K[X_1,X_2]/(X_1X_2)\Big/(X_i)/(X_1X_2)\simeq K[X_1,X_2]/(X_i)\simeq K[X_1]\;\text{or}\;K[X_2],$$ which have dimension $1$. Thus $K[X_1,X_2]/(X_1X_2)$ has dimension $1$.
Bernard
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Actually I have the next problem: Find $X,Y\subset \mathbb{A}^n$ affine varieties s.t. $X\cap Y=\emptyset$ and $dim X+ dim Y\leq n1$. For $n\geq 3$ I found but for $n=2$ or $n=1$ I didn't succeed. Could you help me? – Problemsolving May 05 '17 at 11:01

1You can take an affine curve, say a line, and a point outside of this line (case $n=2$) , or two distinct points (both cases). – Bernard May 05 '17 at 11:18
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$(X_1X_2)=(X_1)\cap(X_2)$ is the primary decomposition
$\dim K[X_1, X_2] = \max\{\dim \frac{K[X_1,X_2]}{(X_1)}, \dim \frac{K[X_1,X_2]}{(X_2)}\}=1$.
user26857
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Problemsolving
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