$A$ is a ring and $I$ is an ideal of $A$ with $I \neq A$. Then, are (1) and (2) equivalent?

Question

$A$: commutative ring with unit.

(1) $I$ is a primary ideal of $A$.

(2) If $a,b \in A$, and $ab \in I$, then there exists a positive integer $n$ such that

$a^n \in I$ or $b^n \in I$.

I notice that they are equivalent when $I$ is prime. So I am looking for a primary ideal which is not prime as a counterexample.

Answer 1

No these are not equivalent. Consider:

$\ \ \ \ $ $(2)'$ $\sqrt{I}$ is prime

and check that $(2)'$ is equivalent to $(2)$.

So for a counterexample, we want a ring $R$ with an ideal $I$ such that $\sqrt{I}$ is prime but $I$ is not primary.

This has been asked about and answered many times on stackexchange, but it is a good exercise to find a counterexample for yourself.