Let R be a right semisimple ring. For x, y ∈ R, show that Rx =Ry iff x = uy for some unit u ∈ U(R).
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 RR= yR ⊕ A = xR ⊕ B, where A, B are right ideals. By considering the composition factors of RR, yRand xR, we see that A ∼B as right R-modules. Therefore, f can be extended to an automorphism g of RR. Letting u = g(1) ∈U(R), we have x = f(y) = g(y) = g(1y) = uy.