According to Lam, let k be any division ring, V be a right k-vector space, and E=End(V), operating on the left of V. then it says clearly V is faithful simple left E module, so E is left primitive ring.
My problem is with V being simple E-Module, as it is faithful because any endomorphism taking whole V to 0 is clearly 0 operator. but why is V is simple? my guess is, let V has a proper E- sub module(non-zero) ,say W, then for any φ in End(V), φ(W) ⊆ W , which implies End(V) ⊆ End(W), not possible as W is a proper subspace of V.
please correct me and tell me what am i missing here. it could be very well trivial, i know.
The ring $E$ acts transitively on the set on non-zero elements of $V$, so there are no proper on-zero $E$-submodules.