Which is the Krull dimension of $\frac{\mathbb{K}[X_1,X_2]}{(X_1X_2)}$?

Question

Let $\mathbb{K}$ be a field. What is the Krull dimension of $\frac{\mathbb{K}[X_1,X_2]}{(X_1X_2)}$?

Answer 1

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 $1$.

Answer 2

$(X_1X_2)=(X_1)\cap(X_2)$ is the primary decomposition

$\dim K[X_1, X_2] = \max\{\dim \frac{K[X_1,X_2]}{(X_1)}, \dim \frac{K[X_1,X_2]}{(X_2)}\}=1$.