Answer to Question #16895 in Abstract Algebra for Irvin

Question

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

Answer 1

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_R, yRand xR, we see that A ∼ B as right R-modules. Therefore, f can be extended to an automorphism g of R_R. Letting u = g(1) ∈U(R), we have x = f(y) = g(y) = g(1y) = uy.