Answer to Question #17108 in Algebra for john.george.milnor

Question

Let M be a left R-module and E = End(RM). If M is a semisimple R-module, show that E is a semisimple E-module.

Answer 1

Every nonzero element m ofMcan be written as m1 + · · · + mn where each Rmi issimple. We claim that each miE is a simple E-module. Once this isproved, then m is contained in the semisimple E-module miE,and we are done. To show that miE is a simple E-module, itsuffices to check that, for any e ∈ E such that mie is nonzero, mi*eE containsmi. Consider the R-epimorphism ϕ : Rmi → Rmie given by rightmultiplication by e. Since Rmi is simple, ϕ is an isomorphism. Let ψ : Rmie → Rmi be the inverseof ϕ, and extend ψ to an f ∈ E. (We can take f to be zero, forinstance, on an R-module complement of Rmie.) Now mief = (mie)ψ= (mie)ϕ−1 = mi, as desired.