Consider the ring $R=K\left[x,y,\frac{x}{y},\frac{x}{y^2},\frac{x}{y^3},\cdots\right]$ , where $K$ is a field. Show that $R$ is a non-Noetherian ring by showing that the chain $\displaystyle\langle x \rangle \subset \langle \frac{x}{y}\rangle \subset \langle \frac{x}{y^2}\rangle \subset \cdots$ is a non-stabilizing chain.
Suppose that the given chain stabilizes. So , $\displaystyle \langle \frac{x}{y^n}\rangle =\langle \frac{x}{y^{n+k}}\rangle$. Then , $\displaystyle \frac{x}{y^{n+k}}=f.\frac{x}{y^n}\implies \frac{x}{y^n}\left(f-\frac{1}{y^k}\right)=0$. Since $R$ is an integral domain so , $f=1/y^k.$ Now we shall show that $f \not \in R$. I'm stuck here. How can I proceed further ?