Why showing that $\bar{xy} \in \bar{P}^{2}$ but that no power of $\bar{y}$ lies in $\bar{P}^2$ shows that it is a prime ideal?

Question

Here is the question I am trying to understand its solution:

Let $R = \mathbb Q[x,y,z]$ and let bars denote passage to $ \mathbb Q[x,y,z] / (xy - z^2).$ Prove that $\overline{P} = (\bar{x}, \bar{z})$ is a prime ideal. Show that $\overline{xy} \in \overline{P}^2$ but that no power of $\bar{y}$ lies in $\overline{P}^2.$ (This shows $\overline{P}$ is a prime ideal whose square is not a primary ideal).

My question is about the second part of the question:

Why showing that $\bar{xy} \in \bar{P}^{2}$ but that no power of $\bar{y}$ lies in $\bar{P}^2$ shows that it is a prime ideal ? Does not that the definiton required either a $x \in \overline{P}^2$ or $y \in \overline{P}^2$? what exactly is the definition of Prime ideal in case of $\bar{P}^{2}$?

Answer 1

There are two claims being made:

$\bar P$ is prime and

$(\bar P)^2$ is not primary.

Why showing that $\bar{xy} \in \bar{P}^{2}$ but that no power of $\bar{y}$ lies in $\bar{P}^2$ shows that it is a prime ideal ?

The answer is "It doesn't. It shows $(\bar P)^2$ is not primary. You are mixing up the halves of the task. They did not offer a hint on how to show $\bar P$ is prime."