# 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:

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

– 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. Mar 26 '14 at 23:55
• The second definition is not as strong as the first. Mar 27 '14 at 2:54
• The definition of primary ideals is symmetric: see here. Apr 1 '14 at 10:23
• It seems a more accurate version of the title would be "Why is the definition of primary ideals the way it is?" Sep 13 '19 at 16:09

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.