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.
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.
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