If $p$ is a prime number in $Z$, how do you show $\langle p^n \rangle$ is a primary ideal in $Z$ Suppose $ab \in \langle p^n \rangle = I$. How do you show either $a \in I$ or $b^m\in I$. It has been some time since I've studied this and would appreciate if someone can help me recall how the usual argument goes. 
Edit: I think you have to write $I = \langle p \rangle ^n$. Then use that fact that a prime ideal is primary.

The statement $ab\in \langle p^n\rangle$ means that $p^n$ divides $ab$. So $p|ab$. So if $p \nmid a$, then $p$ must divide $b$, i.e. $b^n \in \langle p^n\rangle$. So $\langle p^n\rangle$ is primary!
