# Concerning \$R+(t^2)=k[t]\$ and \$R+(y^2)=k[x,y]\$

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.

$$R=k(x,y+xy^2)$$ 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 $$R subset S$$?