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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