Is the ideal $\langle x, xy, xy^2, \cdots \rangle$ finitely generated in $k[x,y]$?

Question

Is the ideal $\langle x, xy, xy^2, \cdots \rangle$ finitely generated in $k[x,y]$?

This is an example for non noetherian ring. But the example in the book used the subring of $k[x,y]$ $$R=\{a+xg: a\in k, g\in k[x,y]\}$$ so that all polynomials of $R$ will have a factor of $x$ in the non constant terms. Then $\langle x, xy, xy^2, \cdots \rangle$ as an ideal in $R$ is not finitely generated. To see this, argue by contrapositive, take the highest $y$ power appearing in each generator in $\{f_1, \cdots, f_m\}$ and call this number $N$, then $xy^{N+1}\not\in \langle f_1, \cdots, f_m \rangle$.

But what about in the ring $k[x,y]$, is the above ideal finitely generated?

Answer 1

Hint $$<x> \subset \langle x, xy, xy^2, \cdots \rangle$$ and $$xy^n \in <x>$$

P.S. Note that the same argument doesn't hold in $R$ since $y \notin R$.

Answer 2

All polynomial rings $A[X_1,\dots,X_n]$ in a finite number of indeterminates over a noetherian ring $A$ are noetherian. A field is noetherian, as a ring.