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Answer on Abstract Algebra Question for Melvin Henriksen

Question #17176
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.
Expert's answer
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.

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