A big list of examples that a power of a prime ideal is not primary in an algebra of finite type over a field

Question

Let $k$ be a field. Let $A$ be an integral domain which is a $k$-algebra of finite type. I would like to know examples that a power of prime ideal of $A$ is not primary. The more example, the better. In other words, I'm asking a big list of such examples.

Answer 1

Let $A = k[X, Y, Z]/(XY - Z^2)$, where $k[X, Y, Z]$ is a polynomial ring over a field $k$. Let $x, y, z$ be the image of $X, Y, Z$ respectively by the canonical homomorphism $\phi\colon k[X, Y, Z] \rightarrow A$. Then $P = (x, z)$ is a prime ideal by this question. We claim $P^2$ is not primary. Suppose otherwise. Since $xy = z^2 \in P^2$, $x \in P^2$ or $y^n \in P^2$ for some $n \ge 1$. Thus $X \in (X^2, XZ, Z^2, XY - Z^2)$ or $Y^n \in (X^2, XZ, Z^2, XY - Z^2)$. However, clearly this is not the case.