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

Question

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.

thanks ahead

Answer 1

I think i have an idea for a solution:

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