What's the motivation of the definition of primary ideals?

$$xy\in\mathfrak q\:\Rightarrow\:\text{either x\in\mathfrak q or y^n\in\mathfrak q for some n\gt0}.$$

Primary ideals can be regard as the generalization of prime ideals and radical.

But why it's defined like that? It's not symmetry. Why not define like that:

$$xy\in\mathfrak q\:\Rightarrow\:\text{either x^n\in\mathfrak q or y^n\in\mathfrak q for some n\gt0}.$$

• Please elaborate on your question. – user122283 Mar 26 '14 at 23:47
• What is there to elaborate upon? The question is precise in what it's asking, and it seems like a very reasonable (albeit perhaps naïve) question to ask when confronted with the definition at first. – ah11950 Mar 26 '14 at 23:55
• The second definition is not as strong as the first. – rschwieb Mar 27 '14 at 2:54
• The definition of primary ideals is symmetric: see here. – user26857 Apr 1 '14 at 10:23
• The question in the title hasn't been answered yet. – Martin Brandenburg Oct 12 '14 at 14:06

Consider $(x^2,xy)$ in the ring $F[x,y]$ where $F$ is a field. According to the normal definition, it is not primary since it doesn't contain any powers of $y$ and doesn't contain $x$.
However, it does satisfy the second definition. If $ab$ is in $(x^2, xy)$, then $x$ divides one of $a$ or $b$, and then that element's square is in this ideal.