Ideals of a polynomial ring $R[x]$ - example

Question

Can you provide me some examples of an ideal $I$ of a polynomial ring $R[x]$.

I need the example for which the set defined below:

$X=\{x \in R: f(x)=0, \forall f \in I \}$

is empty. I know, that e.g. $f(x) = x^2 + 1$ could belong to such an ideal.

Answer 1

Finally I found non-trivial example:

$I = $ { $f \in R[x] :f | (x^2 + 1) $ }

Then, $X$ defined in my question is empty, since $(x^2 + 1) \in I$, and $(x^2 + 1)$ has no roots in $R$. Thank you very much for discussion in comments.