4. Example 4.9.1 - primary ideals

In $A = Z$ , every primary ideal is of the form $(p^{n})$ or $(0)$ , for $p$ prime and $n \in N$
Let $A = k [x, y]$ , and let $q = (x, y^{2})$ . The radical of this ideal is $(x, y)$ , but we can check that no power of $(x, y)$ is equal to $q$ . Thus not all primary ideals are 1;;powers of prime ideals.
There are also examples of prime powers which are not primary. I think an example would be...