Let $D$ be a field, $V$ an infinite dimension vector space over $D$, put $R=\operatorname{End}(V)$. It is clear that $R$ is a non-commutative Ring. What is the form of simple submodules of $R$? Is there any book study the properties of the ring $R$?
Thanks for any help.
Because the ring is prime, all of its minimal right ideals (simple submodules of $R_R$) have to be mutually isomorphic. (If there were two that weren't isomorphic, the two semisimple components generated by these two isoclasses would have product zero, contradicting the ring's primeness.)
One particular simple minimal ideal is the one generated by a map whose image has dimension $1$ (and the rest are isomorphic.)
Another interesting thing to know is that the right (and left) socle of $R$ is the collection of endomorphisms with finite dimensional images.
The ring you are describing is currently 89% complete in the Database of Ring Theory which should keep you busy with lots of its properties.
Some of the most important properties are that it is right and left primitive, von Neumann regular and right self-injective, but not Dedekind finite and not unit regular.