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.

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.*

extended to an automorphism

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