# Answer to Question #16895 in Abstract Algebra for Irvin

Question #16895

Let R be a right semisimple ring. For x, y ∈ R, show that Rx =Ry iff x = uy for some unit u ∈ U(R).

Expert's answer

If

*x*=*uy*where*u**∈*U(*R*), then*Rx*=*Ruy*=*Ry*. Conversely, assume*Rx*=*Ry*. Then, there exists a right*R*-isomorphism*f*:*yR → xR*such that*f*(*y*) =*x*. Write*R*=_{R}*yR**⊕**A*=*xR**⊕**B,*where*A, B*are right ideals. By considering the composition factors of*R*and_{R}, yR*xR*, we see that*A**∼**B*as right*R*-modules. Therefore,*f*can be extended to an automorphism*g*of*R*. Letting_{R}*u*=*g*(1)*∈*U(*R*), we have*x*=*f*(*y*) =*g*(*y*) =*g*(1*y*) =*uy.*
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