Answer to Question #23465 in Algebra for john.george.milnor

Question

Let A,B be left (resp. right) ideals in a semiprime ring R. Show that AB = 0 iff BA = 0.

Answer 1

It suffices to prove the “only if”part, so assume AB = 0, where A,B are (say) left ideals. We have
(BA)^2 = BABA = 0,
so BA = 0 since R issemiprime.