A primary ideal of a commutative (not necessarily unital) ring $R$ is a non-trivial ideal $Q$ such that if $ab\in Q$ for elements $a,b \in R$ then $a\in Q$ or $b^n \in Q$ for some $n\in \mathbb{N}$.
I don’t understand the proof for this. I have written to look at the ring $\mathbb{Z[x]}/ \langle x^2 \rangle$ and look at the cosets of $f,g,fg$ and $g^2$ and we see that if $fg\in Q$ and $f \notin Q$ then $g^2\in Q$.
Why is this?
It is easy to see that $\langle x,2\rangle$ is a maximal ideal in $\mathbb Z [x].$ Also, $$\langle x,2\rangle^2\subseteq \langle x^2,2\rangle\subseteq \langle x,2\rangle.$$ So $\operatorname{rad}(\langle x^2,2\rangle)=\langle x,2\rangle.$ You can deduce the result using the following fact: in a commutative ring with identity, an ideal whose radical is a maximal ideal is a primary ideal(prove!).