Are the following two ideals equal? How to prove it, or show they are not?

Question

$I= \langle x-y^2, x-y^3, x-y^4,... \rangle, $ and $J=\langle x-y^2, x-y^3\rangle$.

Obviously $J \subset I$, but what about the reverse inclusion?

Answer 1

A nice example for Hilbert Basis. Anyway take $x-y^2$ and $x-y^3$ and subtract them to get $y^3-y^2$, now multiply by $y$ (its an ideal) so you get $y^4-y^3$ subtract this from $x-y^3$ and you get $x-y^4$. The rest should now be clear.

Answer 2

$$K[X,Y]/I\simeq\frac{K[X,Y]/(X-Y^2)}{I/(X-Y^2)}\simeq K[Y]/(Y^2-Y^3,Y^2-Y^4,\dots)=K[Y]/(Y^2-Y^3)$$

$$K[X,Y]/J\simeq\frac{K[X,Y]/(X-Y^2)}{I/(X-Y^2)}\simeq K[Y]/(Y^2-Y^3)$$

But if $R$ is a noetherian ring, and $J\subset I$ are ideals of $R$ such that $R/J\simeq R/I$ then $J=I$.