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Given a polynomial ring $k[x,y]$ and a ideal $I=(f)$ generated by element $f$. Consider the quotient ring $k[x,y]/I$,I have the following questions:

  1. If $f$ is in the form of $g(x)+h(y)$, can the elements of quotient ring be written as $\sum^{n_{1}}_{i=0}\sum^{n_{2}}_{j=0}a_{ij}x^{i}y^{j}$, with $a_{ij}\in k$, $n_{1}< deg(g)$ and $n_{2}<deg(h)$?
  2. If we do not assume the specific form of $f$, can we still write down the explicit form of those elements?(In terms of $x$ and $y$ and degree of $f$)
Ken.Wong
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  • My answer in this question may help: https://math.stackexchange.com/questions/2875659/how-to-prove-a-specific-quotient-of-polynomial-ring-is-a-free-module – Z Wu Jun 13 '21 at 03:54

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No, I do not think the first one is correct, take $\mathbb{R}[x,y]/(y-x^2)$, that is wherever you see $y$ make it $x^2$, then the quotient becomes $\mathbb{R}[x,x^2]\cong \mathbb{R}[x]$, and not $\mathbb{R}[x]/x^2$ like you said.

Locally unskillful
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