Answer to Question #17176 in Abstract Algebra for Melvin Henriksen

Question

For a subset S in a ring R. Let R be a semisimple ring, I be a left ideal and J be a right ideal in R. Show that annl (annr(I)) = I and annr (annl(J)) = J.

Answer 1

By symmetry, it is sufficient to prove the above “Double Annihilator Property” for I. Let I = Re,where e = e2 and let f = 1− e. We claim that annr(I)= fR. Indeed, since I · fR = RefR = 0, we have fR ⊆annr(I). Conversely, if a ∈annr(I), then ea = 0 so a = a − ea ∈ fR. This proves annr(I) = fR,and hence similarly annl (annr(I)) = annl(fR)= Re = I.