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# 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 ofMcan be written as m1 + &middot; &middot; &middot; + 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 &isin; E such that mie is nonzero, mi*eE containsmi. Consider the R-epimorphism ϕ : Rmi &rarr; Rmie given by rightmultiplication by e. Since Rmi is simple, ϕ is an isomorphism. Let &psi; : Rmie &rarr; Rmi be the inverseof ϕ, and extend &psi; to an f &isin; E. (We can take f to be zero, forinstance, on an R-module complement of Rmie.) Now mief = (mie)&psi;= (mie)ϕ&minus;1 = mi, as desired.

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