Associated prime ideals in F[X,Y] Let  $F$ be a field and $R=F[X,Y]$ be the polynomial ring of two variables over $F$. Let  $I\subset R$ be the ideal generated by  $X^2$ and $XY$, find the associated prime ideals of  $R/I$.I'm really stuck on this one, I'm trying to show that if  $P\in Ass(R/I)$ then there is an injective map $i:R/P\rightarrow R/I$ and I'm trying to find such a map. I'm not sure if I'm on the write track or not. Any ideas. Thank you!
 A: Hint: $(x^2, xy) = (x) \cap (x, y)^2$
Read up on primary decompositions.
A: As suggested, the key here is the Primary Decomposition Theorem. Here we cite it for clarity:
$\mathit{Theorem}:$ Let $R$ be a Noetherian ring. Then every proper ideal $I$ in $R$ has a minimal primary decomposition. If $I=\bigcap^{m}_{i=1}Q_{i}=\bigcap^n_{i=1}Q^{\prime}_{i}$ are two minimal primary decompositions for $I$ then the sets of associated primes in the two decompositions are the same: $\left\{\mathrm{rad}\,Q_{1},...,\mathrm{rad}\,Q_{m}\right\}=\left\{\mathrm{rad}\,Q^{\prime}_{1},...,\mathrm{rad}\,Q^{\prime}_{n}\right\}$. Moreover, the primary components $Q_{i}$ belonging to the minimal elements in this set of associated primes are uniquely determined by $I$. 
Now let $I=(x^{2},xy)$ in $R[x,y]$. Then $(x^{2},xy)=(x)\cap (x,y)^{2}=(x) \cap (x^{2},y)$ are two minimal primary decompositions for $I$. The associated primes of $I$ are then $(x)$ and $\mathrm{rad}\,((x,y)^{2})=\mathrm{rad}\,((x^{2},y))=(x,y)$. We note that $(x)$ is an isolated prime since $(x) \subset (x,y)$, and $(x,y)$ is an embedded prime. 
