# Answer to Question #17176 in Abstract Algebra 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*=*e*2 and let*f*= 1*− e*. We claim that ann*r*(*I*)=*fR*. Indeed, since*I · fR*=*RefR*= 0, we have*fR**⊆*ann*r*(*I*). Conversely, if*a**∈*ann*r*(*I*), then*ea*= 0 so*a*=*a − ea**∈**fR*. This proves ann*r*(*I*) =*fR*,and hence similarly ann*l*(ann*r*(*I*)) = ann*l*(*fR*)=*Re*=*I.*
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