# Module homomorphism

1. Aug 29, 2008

### peteryellow

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.

2. Aug 29, 2008

### morphism

Re: algebra

peteryellow, you should know by now that the rules of this forum require you to show us what you've done before we offer any help! So, what have you done?