Answer to Question #16876 in Algebra for sanches

Question

Let A, B be left ideals in a ring R, and any idempotent e ∈ R. Show that
eR ∩ (A + B) = (eR ∩ A) + (eR ∩ B), and eR + (A ∩ B) = (eR + A) ∩ (eR + B).
no longer hold if eR is replaced by Re.

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.