Isomorphism or non-isomorphism of two specific local rings

Question

Let $K$ be a field and set $A=K[X,Y]/(XY)$ and $B=K[X,Y]/(Y^2-X^3-X^2)$.

Are the two local rings $A_{(X,Y)}$ and $B_{(X,Y)}$ isomorphic?

I think that they are non-isomorphic but I can't prove this.

Answer 1

The elements $x,y$ are zero-divisors in the ring $A$ and thus in the local ring $A_{(x,y)}$ too. On the other side $B$ and thus $B_{(x,y)}$ are domains.