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

Question

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

Answer 1

We shall prove: if A is a leftideal, thenannl(A) ⊆annr(A). (This clearly givesthe desired statement, by symmetry.) Indeed, let B = annl(A), which isalso a left ideal. By definition, BA = 0, it is iff AB = 0, that is, B ⊆annr(A). By symmetry we obtain the result.