Primary ideals confusion with definition

Solution 1:

You’ve misunderstood the effect of commutativity. It’s not either ... or: it’s both ... and. Let me repeat the definition:

A proper ideal $Q$ of a commutative ring $A$ is primary if whenever $xy\in Q$, then either $x\in Q$, or $y^n\in Q$ for some $n>0$.

Now suppose that some $xy\in Q$. $A$ is commutative, so $yx\in Q$ as well, and the definition then requires that either $y\in Q$, or $x^n\in Q$ for some $n>0$. In other words, the definition actually implies that $xy\in Q$ iff both

• either $x\in Q$, or $y^n\in Q$ for some $n>0$, and
• either $y\in Q$, or $x^n\in Q$ for some $n>0$.

An equivalent formulation that may be clearer: if $xy\in Q$, and $x,y\notin Q$, then there are integers $m,n>1$ such that $x^m,y^n\in Q$. $P^2$ fails this condition: $\bar x,\bar y\notin P^2$, and no power of $\bar y$ belongs to $P^2$.