Finding a primary decomposition Let $k$ be a field, and $R=k[x,y]$. 
I'm supposed to find two different minimal primary decompositions of the ideal $(x^2y, y^2x)$.
It's easy to see that one minimal primary decomposition is $(x)\cap(y)\cap(x^2, y^2)$.
My question: What's the second minimal primary decomposition of the above ideal?
EDIT: I've now added the minimality requirement to the question.
 A: Since you suggest this is homework, I'll put this in the form of hints:
(1) Remember that the set of associated primes for a primary decomposition is unique. So you are looking for $\mathfrak{p} \cap \mathfrak{q} \cap \mathfrak{r}$ where $\sqrt{\mathfrak{p}} = (x)$, $\sqrt{\mathfrak{p}} = (y)$ and $\sqrt{\mathfrak{r}} = (x,y)$.
(2) Show that $\mathfrak{p}$ and $\mathfrak{q}$ must be $(x)$ and $(y)$. So the place where you have room to play is in choosing $\mathfrak{r}$.
At this point, I have trouble giving good general hints. The next two, which are far more helpful, will be put in ROT13.
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A: I'm a bit late with this but even so:
$$ \begin{align} (x^2 y, y^2 x) &= (x^2 y - y^2 x, y^2 x) \\
 &= (xy(x-y), y^2 x) \\
 &= (y) \cap (x(x-y), y^2x) \\
 &= (y) \cap (x) \cap (x-y, y^2)
\end{align}$$
It remains to be shown that $(x-y, y^2)$ is primary: 
Let $r(I)$ denote the radical of $I$. Then $r((x-y, y^2)) = (x-y, y) = (x,y)$. To see that $(x,y)$ is maximal note that $k[x,y]/(x,y) \cong k$ is a field. We know that if $r(I)$ is maximal then $I$ is primary so we're done.
