I have a problem saying
"Check that $k[[x,y,z]]/(x^3-z^2,y^2-xz,z^3)$ is a Gorenstein ring while $k[[x,y,z]]/(x,y,z)^2$ is not, given $k$ is a field."
Anyway, easily check that Krull dimensions of both rings are 0. And there is a theorem stating "$R$ is Gorenstein iff $l_R(0:m)=1$ given that $(R,m)$ is Noetherian local ring".
I used this theorem and since $$(x,y,z)/(x,y,z)^2=(0:_{k[[x,y,z]]/(x,y,z)^2} (x,y,z)/(x,y,z)^2) \supset (x,y^2,z^2,xy,xz,yz)/(x,y,z)^2 \supset (x,y,z)^2/(x,y,z)^2,$$ the latter is not Gorenstein.
But I cannot use the same technique to prove the former, since unlike the latter, I cannot find $(0:m)$.
So, can anyone help me solve this? Did I choose the right approach?