# Answer to Question #16876 in Algebra for sanches

Question #16876

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.

eR ∩ (A + B) = (eR ∩ A) + (eR ∩ B), and eR + (A ∩ B) = (eR + A) ∩ (eR + B).

no longer hold if eR is replaced by Re.

Expert's answer

If

extended to an automorphism

*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*and_{R}, yR*xR*, we see that*A**∼**B*as right*R*-modules. Therefore,*f*can beextended to an automorphism

*g*of*R*. Letting_{R}*u*=*g*(1)*∈**U(**R*), we have*x*=*f*(*y*) =*g*(*y*) =*g*(1*y*) =*uy.*Need a fast expert's response?

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