<s>[INST] Proving that $(X-1,Y^2-4Y-XY+Y+4)=(X-1,Y-2) $ We consider the ideal $I=(X-1,Y^2-4Y-XY+Y+4)$ of the polynomial ring $k[X,Y]$. I need to prove that $(X-1,Y^2-4Y-XY+Y+4)=(X-1,Y-2) $. Since $Y^2-4Y-XY+Y+4=(Y-2)^2-Y(X-1)$, it is easy to see that $I \subseteq (X-1,Y-2) $, but I can't prove that $(X-1,Y-2) \subseteq I$.
Any help would be appreciated.
 [/INST] The claim is false since
$$\begin{align}
(x-1,f(x)) &= (x-1,\,f(x)\bmod x-1)\\
           &= (x-1,\,f(1))\\
           &= (x-1,\,y^2-4y+4)\ \ {\rm in\ OP}\\
           &= (x-1,\,(y-2)^2)
\end{align}\qquad$$
but $\,(x\!-\!1,(y\!-\!2)^2)\neq (x\!-\!1,y\!-\!2),\,$ else $\,y\!-\!2 = (x\!-\!1)f + (y\!-\!2)^2g,\,$ so $\,y\!-\!2\mid f,\,$ so cancelling $y\!-\!2$ yields $\,1 = (x\!-\!1)f'+(y\!-\!2)g,\,$ so $\,1 = 0\,$ by eval at $\,x=1,y=2$.
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