Krull dimension of affine $\Bbbk$-algebra

Question

Given an ideal, $\mathfrak{a} = \langle x_2x_3 \rangle \subseteq \Bbbk[x_1, x_2,x_3]$, where $\Bbbk$ is a field. We have that the maximal set of indeterminates independent modulo the ideal $\mathfrak{a}$, $\{x_1,x_2\}$, is of cardinality 2. Therefore the dimension of $\Bbbk[x_1,x_2,x_3]/\mathfrak{a}$ is 2. I am not able to find a maximal chain of prime ideals of length 2. Can someone help me construct it?

Answer 1

What about $(x_3)/\mathfrak a\subsetneq(x_2,x_3)/\mathfrak a\subsetneq(x_1,x_2,x_3)/\mathfrak a$?