TITLE: Two (other) rings...are they isomorphic?
QUESTION [21 upvotes]: 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?), 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.

REPLY [10 votes]: 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!