Let $k$ be a field, $A = k[X_1,X_2,...]$, $\mathfrak{p} = (X_1,X_2,...)$, $I = (X_1^2-X_1,X_2^2-X_2,...)$, $M= A/I$. I am trying to show that $\operatorname{Ass}_{A_\mathfrak{p}}(M_\mathfrak{p}) = \{ \mathfrak{p}A_\mathfrak{p}\} $, the associated primes of $M_\mathfrak{p}$.
I have shown that $M_\mathfrak{p} \cong k$. I know $\mathfrak{p}A_\mathfrak{p}$ to be prime and maximal amongst ideals, so if it is an annihilator, it is an associated prime. The problem is, of what could it be an annihilator? When identifying $M_\mathfrak{p}$ with $k$, neither 0 nor 1 seem to do the trick, and other options seem arbitrary.
I would like a hint how to show that indeed $\mathfrak{p}A_\mathfrak{p}$ is associated, perhaps I can then work out the fact that it is the only one by myself.
Hint. Just prove that $IA_P=PA_P$.