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*_{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*(1*y*) = *uy.*

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