# $\mathbb{K}[[x,y]]/\langle\, f\,\rangle \cong \mathbb{K}[[x]],\mathbb{K}[[y]]$. in formal power series ring

Let $$\mathbb{K}[[x,y]]$$ be the ring of formal power series in $$x,y$$. Now let $$f(x,y) = \sum_{i+j \geq 1} a_{i,j}x^iy^j \in \mathbb{K}[[x,y]]$$ so that $$a_{1,0}$$ or $$a_{0,1}$$ is not zero.

I would like to prove that $$\mathbb{K}[[x,y]]/\langle\,f\,\rangle\cong \mathbb{K}[[x]],\mathbb{K}[[y]]$$.

My ideas so far: I should find a map (an isomorphism) that maps any factor in $$\langle\,f\,\rangle$$ to 0, for example for $$f(x,y) = y$$ it's obvious what the map should be. I'm having difficulties see what the map should be in the general case.

every $$g(x,y)\in \mathbb{K}[[x,y]]$$ you can write uniquely as $$g(x,y) = h(x,y)+h_0(x)$$ when $$h(x,y)\in ,h_0(x)\in \mathbb{K}[[x]]$$ then the homomorphism $$g \to h_0$$ gives what we wanted (its kernel is $$$$ and image is $$\mathbb{K}[[x]]$$).