Consider the following rings:


$A=\mathbb{C}\lbrace x,y,u \rbrace /(xy+x^3,y^2,xy^2+x^5) \ $ and 


$B=\mathbb{C}\lbrace x,y,u \rbrace /(xy+x^3,y^2+ux^4,xy^2+x^5)$


There is an isomorphism of $\mathbb{C}$-algebras between $A$ and $B$?


Here $\mathbb{C}\lbrace x,y,u \rbrace$ denotes the formal series ring. I'm trying to prove that there is an isomorphism between $A$ and $B$, but I suspect that this isomorphism can not exist. Furthermore, I would like to see if there is at least one morphism between $A$ and $B$ or maybe between $B$ and $A$?


I would be happy with any suggestions.



These rings are isomorphic?
Set $z=y+x^2$. Then 
$$
 (xy+x^3,y^2,xy^2+x^5)=(xz,(z-x^2)^2,xz(y-x^2)+2x^5)
 =(xz,x^5,z^2+x^4)
$$
and similarly $(xy+x^3,y^2+ux^4,xy^2+x^5)=(xz,x^5,z^2+(1+u)x^4)$. So, after setting $x'=x\root4\of{1+u}$ we get the required isomorphism.


