Proving Primary Ideals in \mathbb{Z}: Peter's Challenge

[Math Amateur]

In Dummit and Foote on page 682 Example 1 reads as follows:
----------------------------------------------------------------------------------------------------------------------------The primary ideals in $\mathbb{Z}$ are $0$ and the ideals $(p^m)$ for $p$ a prime and $m \ge 1$.-----------------------------------------------------------------------------------------------------------------------------

So given what D&F say, (4) is obviously not primary.

I began trying to show from definition that (4) was not a primary from the definition, but failed to do this

Can anyone help in this ... and come up with an easy way to show that (4) is not primary?

Further, can anyone please help me prove that the primary ideals in $\mathbb{Z}$ are 0 and the ideals $(p^m)$ for p a prime and $m \ge 1$.

PeterNote: the definition of a primary idea is given in D&F as follows:

Definition. A proper ideal Q in the commutative ring R is called primary if whenever $ab \in Q$ and $a \notin Q$ then $b^n \in Q$ for some positive integer n.

Equivalently, if $ab \in Q$ and $a \notin Q$ then $b \in rad \ Q$

 

[economicsnerd]

$(4)$ is primary, and this agrees with what D&F said. Indeed $(4)=(p^m)$ where, $p=2$ and $m=2$.

 

[Math Amateur]

Thanks economicsnerd!

Not entirely sure how I missed that ... :-(