Given the following primary decomposition in $k[x,y]$, $k$ field:
$$I=\langle xy,x^3-x^2\rangle=\langle x\rangle \cap \langle x-1,y\rangle \cap \langle x^2,y\rangle,$$
I want to compute the minimal primes of $k[x,y]/I$.
The associated primes are:
$\sqrt{\langle x \rangle}=\langle x \rangle$ (prime);
$\sqrt{\langle x-1,y \rangle}=\langle x-1,y \rangle$ (maximal).
$\sqrt{\langle x^2,y \rangle}=\langle x,y \rangle$, because of the inclusion $\supseteq$ is obvious and $\langle x,y\rangle$ is maximal.
So, the minimal primes of $k[x,y]/I$ are given by $\langle x-1,y \rangle/I$ and $\langle x\rangle/I$.
Once I was beginner and the corectness of the answer is very important to me, I'd like to know if I am correct.
Many thanks in advance!