Let $k$ be a field of characteristic zero.

Let $R subseteq k(t)$ be a $k$-subalgebra of $k(t)$

such that:

**(i)** $(t^2) not subseteq R$.

**(ii)** $R+(t^2)=k(t)$.

We have $t=r+st^2$ for some $r in R, s in k(t)$, and then

$R ni r = t-st^2$, so there exists an element $r$ in $R$ of the form

$t-st^2$; however, it is not clear if such $r$ must be a generator of such algebra $R$.

Question 1:

Is one of the generators of $R$ must be of the form $lambda t + f(t)$,

where $lambda in k$ and $f(t) in k(t)-{0}$?

For example $k(t+t^2,t+t^3)$ (here $lambda=1$).

More generally, same question for $k(x,y)$, namely:

Let $R subseteq k(x,y)$ be a $k$-subalgebra of $k(x,y)$

such that:

**(i)** $(y^2) not subseteq R$.

**(ii)** $R+(y^2)=k(x,y)$.

Question 2:

Is one of the generators of $R$ must be of the form $lambda x + f(y)$,

where $lambda in k$ and $f(y) in k(y)-{0}$?

For example $k(x,y+y^2,y+y^3)$ (here $lambda=0$).

**Remarks:**

**(a)** There is no assumption on the number of generators of $R$ in both question 1 and question 2.

**(b)** Also asked here.

Edit:

$R=k(x,y+xy^2)$ is a counterexample to question 2..

So my new questions 1 and 2 are: What can be said about such $R$ or about the extension $R subset S$?

Thank you very much!