Affine variety example.

Question

I know how to show the statement. But I cannot find an example (the part I underlined by a yellow pen) please help me for finding an example.

Note: For example, can I consider the following example; the irreducible cubic plane curve $Z(y^2 − x^3 + x)$ ?

If this true, please explain a little bit?:) thanks:)

Answer 1

a) The simplest example is with $n=r=1$: take $\mathfrak a=\langle X_1(X_1-1)\rangle \subset k[X_1]$.
Then $Z(\mathfrak a)=\{0,1\}$ has two irreducible components $\{0\}, \{1\}$ each of dimension $n-r=1-1=0$.

b) If that was too degenerate for your taste, the next simplest example is with $n=2,r=1$ and $\mathfrak a=\langle X_1X_2\rangle \subset k[X_1,X_2]$.
Then $Z(\mathfrak a)=Z(X_1)\cup Z(X_2)$ has two irreducible components $Z(X_1), Z(X_2)$, each of dimension $n-r=2-1=1$.