# 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

By the theorem on Reduction to Echelon Forms, we can find invertible

where

*R*= M*n*(*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*^{−}^{1}*c*^{−}^{1})(*cdc*^{−}^{1}) =*ue,*where

*u*: =*b*^{−}^{1}*c*^{−}^{1}is a unit and*e*: =*cdc*^{−}^{1}is an idempotent.
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