Let $M_R$ be any module. Recall that $M_R$ is called quasi-injective if every $R$-homomorphism $N\to M$ from a submodule $N$ of $M$ extends to $M$, $M_R$ is called quasi-continuous if it's both CS (i.e., every submodule is essential in a summand) and $C3$ (the sum of two summands with zero intersection is also a summand). Recall that every module has a quasi-injective hull and a quasi-continuous hull in $E(M)$. (i.e., a minimal quasi-injective and a minimal quasi-continuous extensions of $M$ in $E(M)$)
Let $F$ be a field with an isomorphism $x \longmapsto \overline{x}$ from $F$ to a subfield $\overline{F} \neq F$. Let $R$ denote the left $F$-space on basis $\lbrace 1,c \rbrace$ where $c^2=0$ and $cx=\overline{x}c$ for all $x\in F$. By straightforward calculations, one deduces that $R$ is a local ring. Also $R_R$ is not CS.
What is the injective hull of $R_R$. Can we compute the quasi-continuous (respectively, the quasi-injective) hull of $R_R$ ?.
Thanks for help.