Ring $R_{ 38 }$

Description:

Let $\sigma:k\to k$ be a field endomorphism of a countable field $k$ such that $\infty > n=[k:\sigma(k)]>1$. $k[x;\sigma]$ is the twisted polynomial ring where $xa:=\sigma(a)x$ for all $a$ in $k$. The ring is $k[x;\sigma]/(x^2)$.

Notes: Composition length finite but different on both sides

Keywords quotient ring twisted (skew) polynomial ring