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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.