I don't understand the "simmetry" of the definition: I is primary if $ab \in I$ then $a \in I$ or $b \in \sqrt{I}$. This definition $a,b$ are completely interchangeable so my head reads it like $ab \in I$ then $a \in I$ or $b \in \sqrt{I}$ or $a \in \sqrt{I}$ or $b \in I$, which probably doesn't make sense since $a$ is not distinguable from $b$ from the definition, so when you have in general that something is primary, one can assume that if $a \not\in I$ then I will prove that $b \in \sqrt{I}$. If I find that $b \not\in \sqrt{I}$ than $I$ is not primary. Correct if I'm wrong or suggest me if there is a better way of thinking about it.
I'd like to prove that a prime radical doesn't imply that the ideal is primary, for example $(xy,y^2)$ in $\mathbb{C}[x,y]$. My question in this context is, why can I affirm that $(xy,y^2)$ simply by noting that $y \not\in (xy,y^2)$ and $x^n \not\in \sqrt{(xy,y^2)} = (y)?$ I assumed that in a specific setting I should verify "both". In fact, $x \not\in (xy,y^2)$ but $y \in \sqrt{(xy,y^2)}$.
Any help would be appreciated.