# Answer to Question #17103 in Algebra for john.george.milnor

Question #17103

Let R be a semisimple ring. Show that any quotient ring of R is semisimple.

Expert's answer

Write

*R*=*R*_{1}*×· · ·×R*where the_{r}*Ri*’s are the simple components of*R*. Consider any quotient ring*R/I*of*R*. After a reindexing, we may assume that*I*=*R*_{1}*⊕**· · ·**⊕**R*for some_{s}*s ≤ r*. Therefore,*R/I**∼**R*_{s}_{+1}*×· · ·×R*is a semisimple ring._{r }
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