# Primary ideals in Z

• Nov 10th 2013, 08:05 PM
Bernhard
Primary ideals in Z
In Dummit and Foote on page 682 Example 1 reads as follows:
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The primary ideals in $\mathbb{Z}$ are 0 and the ideals $(p^m)$ for p a prime and $m \ge 1$.

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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 $m \ge 1$.

Peter

Note: 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 $b \in rad \ Q$
• Nov 26th 2013, 08:43 PM
Haven
Re: Primary ideals in Z
(4) is primary, as $4=2^2$ and 2 is prime. Suppose $ab \in (4)$. Then $4|ab$. Suppose $a \not\in (4)$, so $4\nmid a$.

But we can have two cases, either $2|a$ or $2\nmid a$:

If $2|a$, as $4|ab$ and $4\nmid a$, we get $2|b$, which means $b\in(2)$ therefore $b^2 \in (4)$.

If $2\nmid a$, then $4|b$, and so $b\in 4$.

You can use a similar proof to prove that the primary ideals are (0) and $(p^m)$.