Quotient Rings of Rings in Several Variables

Question

How should I interpret $$\left(\left(\mathbb{Z[x,y]}/(x) \right)/(y) \right)?$$ Is $$\left(\left(\mathbb{Z[x,y]}/(x) \right)/(y) \right) \cong \left(\left(\mathbb{Z[x,y]}/(y) \right)/(x) \right)?$$

Furthermore, if $f(x,y) \in \mathbb{Z[x,y]}$, what is the representative of $f$ in $\left(\left(\mathbb{Z[x,y]}/(x) \right)/(y) \right)?$

Answer 1

Elements of $\left(\left(\mathbb{Z[x,y]}/(x) \right)/(y) \right)$ are cosets of cosets. Explaining by examples-

$\textbf{Example 1-}$ Take $f(x,y)$= $x+y$ then in $\left(\left(\mathbb{Z[x,y]}/(x) \right)/(y) \right)$ , $[f(x,y)+(x)]+(y)=y+(y)=0$

$\textbf{Example 2-}$ Take $f(x,y)$=$x^2+xy+y+7$ , then is image will be $[f(x,y)+(x)]+(y)=y+7+(y)=7$, ( NOTE- $xy \in (x)$ & $(y)$ both)

Answer 2

Yes, the rings are isomorphic. To prove this, define a surjective morphism from $\mathbf{Z}[x,y]$ to each of the rings, and show that the kernels of the two morphism are identical.

For the second question, it will be the image of $f$ under the surjective morphism.

Answer 3

$\mathbb{Z}[x,y]/(x)/(y) \simeq \mathbb{Z}[x,y]/(x,y) \simeq \mathbb{Z}$ under the map $f\mapsto f(0,0)$.

Yes, isomorphic.