Answer to Question #16875 in Algebra for sanches

Question

Let A, B be left ideals in a ring R. Show that for any idempotent e ∈ R, we have the following:
eR + (A ∩ B) = (eR + A) ∩ (eR + B).

Answer 1

Here, we need only prove the inclusion “⊇”. For any element x ∈ (eR +A) ∩ (eR + B), we can write
x = er + a = es + b, where r, s ∈ R, a ∈ A, and b ∈ B. Then ex = er + ea = x − a + ea, so x − ex ∈ A. Similarly, x − ex ∈ B. Thus, x = ex + (x − ex) ∈ eR + (A ∩ B), as desired.