Consider the local rings


$$R = \mathbb{C}[[x,y,z,w]]/\langle xyz+xyw+xzw+yzw\rangle$$


and


$$S = \mathbb{C}[[x,y,z,w]]/\langle xyz+xyw+xzw+yzw+xyzw\rangle.$$


Is $R$ isomorphic to $S$?


**Some context**: I am trying to understand formal neighborhoods of points on certain varieties. I expect one answer, and I'm getting a different answer. This is the first nontrivial case where the answer that I get does not obviously agree with the answer that I expect.


**Some history**: In a previous post ([Two rings...are they isomorphic?](https://mathoverflow.net/questions/186886/two-rings-are-they-isomorphic)), I asked a version of this question with one fewer variable, and Bjorn Poonen pointed out that the rings are isomorphic because there is only one kind of rational double point. I think this was essentially an accident, hence the new post.



Two (other) rings...are they isomorphic?
Consider the map $\varphi:S\to R$ given by
$$\varphi(x) = \frac{4x}{4-x},$$
and similarly for $y$, $z$, and $w$. One needs to check that this is well-defined;
indeed, $\varphi$ maps $xyz+xyw+xzw+yzw+xyzw$ to 
$$\frac{4^4(xyz+xyw+xzw+yzw)}{(4-x)(4-y)(4-z)(4-w)}.$$
Since $\varphi$ obviously induces an isomorphism on the associated graded of the filtration by powers of the maximal ideal, it is itself an isomorphism.  

More explicitly, the inverse homomorphism $\psi:R\to S$ is given by
$$\psi(x) = \frac{4x}{4+x},$$
and similarly for $y$, $z$, and $w$.


Note also that this generalizes to any number of variables.


Thanks David and Vladimir for your answers; that was really helpful for getting me going!


