Let $k$ be a division ring and $V$ be a left $k$-vector space of infinite dimension. Let $E=\mathrm{End}(V)$, defined as a ring of right operators on $V$. My questions are:
(1) Why $V$ is a simple right $E$-module?
(2) If $v∈V$ and $e∈E$ are such that $v∉Ve$, what is the "suitable" basis by which we could define an $x∈E$ that vanishes on $Ve$ but not on $v$?
Thanks for any help!
Hints
(1) Let $v\in V$, $v\ne0$; then, for each $w\in V$ there is $f\in\operatorname{End}(V)$ such that $vf=w$. Thus $vE=V$, which means that the only proper $E$-submodule of $V$ is $\{0\}$.
(2) If $B$ is a basis for $Ve$, then $B\cup\{v\}$ is linearly independent and so it can be completed to a basis for $V$.