A cyclic right ideal which is not finitely generated

Question

I am looking for a two-sided ideal $I$ in a ring with identity such $I$ is not finitely generated as a left ideal but it is cyclic as a right ideal.

Answer 1

Hint:

Let $s:F\to F$ be a field endomorphism such that $[F:s(F)]=\infty$.

The twisted polynomial ring $F[x;s]$ is like the ordinary one, except the indeterminate $x$ is not assumed to commute with elements of $F$. Instead, $xa=s(a)x$ for all $a\in F$.

This ring $F[x;s]/(x^2) $ contains an example of what you're looking for: take a look!