Let $Y,Z$ be affine varieties (irreducible). Then every irreducible components $W$ of $Y \cap Z$ correspends to the minimal prime ideals $\frak{p}$ of the principal ideal $(f)$. Why can we say that it is "minimal" ideal?
Because $Y \subset Z$ we obtain $Y \cap Z \subset Z$. Thus, for any irreducible component $W_i$ of $Y\cap Z$, we get $ W_i \subset Y \cap Z \subset Z$.
$I(W_i) \supset I(Y \cap Z) \supset I(Z)=\sqrt{(f)}=(f)$.
Thus
$I(W_i) \supset (f)$.
But I cannot understand where the word "minimal" appears?
In the following proof of the statement(Hartshorne page 48 Prop 7.1), the minimal prime ideal is used, but I cannot understand how it is used.
Prop Let $YU,Z$ be varieties of dimension $r,s$ in $\mathbb{A}^n$. Then every irreducible component $W$ of $Y\cap Z$ has dimension $\geq
r+s-n$.PROOF. We proceed in several steps. First, suppose that $Z$ is a hyper
surface, defined by an equation $f=0$. If $Y \subset Z$, there is
nothing to prove. If $Y \nsubseteq Z$, we must show that each
irreducible component $W$ of $Y\cap Z$ has dimension $r-1$. Let $A(Y)$
be the affine coordinate ring of $Y$. Then the irreducible components
of $Y\cap Z$ correspond to the minimal prime ideals $\frak{p}$ of the
principal ideal $(f)$ in $A(Y)$. Now by Krull's Hauptidealsatz, each
such $\frak{p}$ has height one, so by the dimension theorem,
$A(Y)/\frak{p}$ has dimension $r-1$. Because the dimesnion of $W$ is
the dimension of its affine coordinate ring, $W$ has dimesnion $r-1$.
PROOF is continued for $Z$ is not a hyper surface case.
Best Answer
This is just the inclusion-reversing bijection between ideals and varieties. Under the dictionary between the ideal side and the variety side, an irreducible subvariety corresponds to a prime ideal cutting it out, and a containment of varieties $X\subset Y$ corresponds to a containment of ideals $I_X\supset I_Y$. As an irreducible component is maximal with respect to containment among irreducible subvarieties, when we apply this correspondence, it gets transformed in to a prime ideal minimal with respect to containment among prime ideals, aka a minimal prime.