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Answer to Question #17116 in Abstract Algebra for sanches

Question #17116
Let R be any semisimple ring. If a ∈ R is such that I = aR is an ideal in R, then I = Ra.
Expert's answer
By the Wedderburn-Artin Theorem, we are reduced to the case when R = Mn(D) where D is a division ring.
Since R = Mn(D) is now a simple ring, we have either I = 0 or I = R. If I = 0, then a = 0 and Ra = I holds. If I = R, then ab = 1 for some b ∈ R. By Dedekind-finiteness of R, ba = 1, so Ra = I also holds.

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