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

Question #17108

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.

Expert's answer

Every nonzero element

*m*of*M*can be written as*m*1 +*· · ·*+*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*contains*mi*. 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.
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