It is known the following equivalence: Let $Y \subset \mathbb{A}^{n}$ be an algebraic set. Then $Y$ is irreducible if and only if $I(Y)$ is prime, where:
$I(Y) = \{f \in K[x_{1},x_{2},..,x_{n}]: \textrm{for all p in Y}, \ f(p)=0 \}$.
So as an example, the author considers $J = \langle (xz-y^2,x^3-yz) \rangle \subset k[x,y,z]$. Then let $Y=V(J)$ (the locus set).
So if we want to find whether $Y$ is irreducible we must study $I(Y)=I(V(J))$ and show this ideal is prime (or show it is not).
The author then shows that $J$ is not a prime ideal. But why is this? don't we need to show that $I(V(J))$ is a prime ideal? why we must check $J$? or do we always have $J=I(V(J))$?