**Edit**: I have reverted my question to its original version (which Bjorn Pooenen answered correctly) as requested in the comments.


Consider the local rings


$$R = \mathbb{C}[[x,y,z]]/\langle xy+xz+yz\rangle$$


and


$$S = \mathbb{C}[[x,y,z]]/\langle xy+xz+yz+xyz\rangle.$$


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.



Two rings...are they isomorphic?
Yes, they are isomorphic.


More generally, if $k$ is any algebraically closed field of characteristic not $2$, and $n$ is given, then all $k$-algebras of the form $k[[x\_1,\ldots,x\_n]]/(f\_2+f\_3+\cdots)$, where each $f\_i$ is homogeneous of degree $i$, and $f\_2$ is a nondegenerate quadratic form, are isomorphic. (I.e., there is only one kind of ordinary double point.)


You can construct an isomorphism to $k[[x\_1,\ldots,x\_n]]/(x\_1^2+\cdots+x\_n^2)$ with your bare hands as follows. First diagonalize the quadratic form to assume that $f\_2=x\_1^2+\cdots+x\_n^2$. Let $m$ be the lowest degree monomial of degree greater than $2$. Then $m$ is divisible by some $x\_i$, say $m=x\_1 g$. Performing the analytic change of variable $x\_1 \mapsto x\_1-g/2$ eliminates $m$ at the expense of introducing new terms of even higher degree. By iterating, one can eventually eliminate all monomials of degree 3 to obtain $f\_3=0$, and then $f\_4=0$, etc. The partial compositions of this sequence of analytic coordinate changes converge to a single analytic coordinate change because they stabilize modulo any given power of the maximal ideal.


