# Module homomorphism

Let V be a countable dimensional vectorspace over a field F .
Let R denote End_F V .
Prove that V is a simple R-module. If $e1 , e2 , . . .$ is a basis
of V , then we have a module homomorphism φ_j from R to V ,
sending f in R to f (e_j ).
Find the ker(φ_j) .
Find Jac(R). here I mean jacobson radical.
Prove that there is exactly one non-trivial twosided ideal,
namely
${f ∈ R| dimf (V ) < ∞}$ Prove that R is isomorphic to R + R here + means direct sum . thanks.