# Answer to Question #17104 in Algebra for john.george.milnor

Question #17104

Let R be a semisimple ring. Show that a simple artinian ring S is isomorphic to a simple component of R iff there is a surjective ring homomorphism from R onto S.

Expert's answer

If

*S**∼**Ri*, we can find a surjective ring homomorphism from*R*to*S*by utilizing the*i*th projection of*R*=*R*1*×· · ·×Rr*. Conversely, suppose*ϕ**:**R → S*is a surjective ring homomorphism. After a reindexing, we may assume that ker(*ϕ*) =*R*_{1}*×· · ·×R*_{r−}_{1}*.*Therefore,*S**∼**R/*ker(*ϕ*)*∼**R*._{r}
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