# Answer to Question #23466 in Algebra for john.george.milnor

Question #23466

Let A ideal in a semiprime ring R. Show that annr(A) = annl(A).

Expert's answer

We shall prove: if A is a leftideal, thenann

*l*(A)*⊆*ann*r*(A). (This clearly givesthe desired statement, by symmetry.) Indeed, let B = ann*l*(A), which isalso a left ideal. By definition, BA = 0, it is iff AB = 0, that is, B*⊆*ann*r*(A). By symmetry we obtain the result.
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