Question

Let R be any semisimple ring. Every element a ∈ R can be written as a unit times an idempotent.

Answer 1

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.