Recently I am working on algebraic geometry. In dealing with a problem, I need the following:
If $\alpha$ is an ideal generated by $r$ elements of the ring $k[x_1,\cdots,x_n]$, suppose $Y$ is an irreducible component of $Z(\alpha)$, then $I(Y)$ is also generated by $r$ elements of $k[x_1,\cdots,x_n]$.
Here $Z(\alpha)$ is the set of points in an affine space that vanish with the evaluation by all elements in $\alpha$ and $I(Y)$ is an ideal of $k[x_1,\cdots,x_n]$ that vanish at all points of $Y$. By irreducible I mean cannot be further decomposed into distinct closed subsets in terms of Zariski topology.
Is there any way to prove or disprove it? In case it is not correct, is there any related proposition?