64 693
Assignments Done
99,3%
Successfully Done
In September 2018

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 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.

Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be first!

Leave a comment

Ask Your question

Submit
Privacy policy Terms and Conditions