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.
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Expert's answer
2012-10-25T10:28:23-0400
If S ∼Ri, we can find a surjective ring homomorphism from R to S by utilizing the ith projection of R = R1 ×· · ·×Rr. Conversely, suppose ϕ: R → S is a surjective ring homomorphism. After a reindexing, we may assume that ker(ϕ) = R1×· · ·×Rr−1. Therefore, S ∼R/ ker(ϕ) ∼Rr.
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