On Herstein's conjecture (Q1204649)

Let \(R\) be a left noetherian ring with identity element and \(A\), \(B\) are left ideals of \(R\) such that \(B\subseteq A\) and for any element \(x\in A\) there exists an integer \(n = n(x) \geq 1\) such that \(x^ n \in B\). The author proves that \(A^ m \subseteq B\) for some integer \(m \geq 1\) if either \(B^{k+1} = B^ k\) for some \(k \geq 1\) or \(B + P/P\) is a left artinian \(R/P\)-module (where \(P\) is the lower nil radical of \(R\)).