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Answer to Question #17117 in Abstract Algebra for sanches

Question #17117
Let R be any semisimple ring. Every element a ∈ R can be written as a unit times an idempotent.
Expert's answer
By the Wedderburn-Artin Theorem, we are reduced to the case when R = Mn(D) where D is a division ring.
By the theorem on Reduction to Echelon Forms, we can find invertible n × n matrices b, c ∈ R such that
d := bac = diag(1, . . . , 1, 0, . . . , 0) (an idempotent). We have now a = (b−1c−1)(cdc−1) = ue,
where u : = b−1c−1 is a unit and e : = cdc−1 is an idempotent.

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