Let $k$ be a field of characteristic zero, and let $R_i$ be a descending chain of $k$-subalgebras of $R_0:=k[x,y]$. Assume that the Krull dimension of each $R_i$ is $2$.
I am looking for an example of such $R_i$'s such that, in addition, $\cap R_i$ is of Krull dimension exactly $1$.
Please see also this question; notice that $R_1=k[x^2,y^2]$, $R_2=k[x^4,y^4]$, $R_3=k[x^8,y^8]$, namely, $R_i=k[x^{2^i},y^{2^i}]$, $i \in \mathbb{N}$, is not an answer to my current question, since $\cap_{i \in \mathbb{N}}k[x^{2^i},y^{2^i}]=k$, which is of Krull dimension $0$.
Thank you very much!