Example of a primary ideal that it's not prime

Question

in the course of Algebra I studied the primary ideals, an ideals $I$ of a commutative ring with identy is called primary if $ab \in I$ and $a\notin I$ implies that $ \exists n \in \mathbb{Z}$ such that $b^n \in I$. It is evident that prime implies primary, I'm looking for an example that shows that the opposite is not true.

Take your favorite ring (which of course is $\mathbb Z$) and try out ideals in it. — Wojowu, Jan 10 '19 at 18:39
See the second line in the wikipedia article. At this site, see this duplicate. — Dietrich Burde, Jan 10 '19 at 19:30

score 3 · Accepted Answer · answered Jan 10 '19 at 18:41

3

Hint:

$p\cdot p\in\langle p^2\rangle$

answered Jan 10 '19 at 18:41

ajotatxe

65,084

score 1 · Answer 2 · answered Jan 10 '19 at 18:42

1

This is the graphic at wikipedia's prime ideal page currently. I thought it also appeared on the primary ideal page, but it looks like it doesn't:

Any patterns present themselves? You might try proving a conjecture...

answered Jan 10 '19 at 18:42

rschwieb

153,510

Example of a primary ideal that it's not prime

2 Answers2