# Answer to Question #16875 in Algebra for sanches

Question #16875

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).

eR + (A ∩ B) = (eR + A) ∩ (eR + B).

Expert's answer

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.Need a fast expert's response?

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