Minimal primes of $\Bbb C[w,x,y,z]/(w^2x,xy,yz^2)$ and their annihilators I am examining the minimal primes of  $R=\Bbb C[w,x,y,z]/(w^2x,xy,yz^2)$, intersections between them, and the annihilators of each minimal prime.
If a minimal prime does not contain x, then it must contain both w and y. If it contains x, then it must contain one of y or z. So I think a complete set of minimal primes is given by $(w,y), (x,y)$ and $(x,z)$. (Good so far I hope?)
The problem is that I second-guess my judgement on conclusions about the annihilators and intersections.
Is it safe to conclude something about the generators of the intersections in terms of the generators of the ideals? The only thing I know for sure is that $(wx, yz)$ is nilpotent and hence contained in all three minimal primes.
I can also see $(wx)\subseteq ann(w,y)$,  $(w^2z^2)\subseteq ann(x,y)$ and $(yz)\subseteq ann(x,z)$, but I lose my train of thought trying to verify that they are equalities. 
Any tips or theorems for clarifying how to think about these items are appreciated. 

$$(w^2x,xy,yz^2)=(\underline{w}^2,xy,yz^2)\cap(\underline{x},xy,yz^2)=(w^2,\underline{x},yz^2)\cap(w^2,\underline{y},yz^2)\cap(x,yz^2)=(w^2,x,y)\cap(w^2,x,z^2)\cap(w^2,y)\cap(x,y)\cap(x,z^2)=(w^2,y)\cap(x,z^2)\cap(x,y)$$ is a reduced primary decomposition, so the associated (minimal) prime ideals are $(y,w),(x,z),(x,y)$.
For intersections, let's consider $(y,w)\cap(x,z)=(xw,zw,xy,yz)$. In $R$ we have $xy=0$, so one can get rid of this.
$Ann_R(x,y)=(z^2w^2)$, $Ann_R(x,z)=(yz)$, and $Ann_R(y,w)=(xw)$.
