Relative nilpotency of left ideals (Q1803078)

Let \(R\) be an associative ring, \(L\) and \(Z\) be left ideals of \(R\) and let \(Z \subseteq \bigcap_{m\in\mathbb{N}}L^ m\). Suppose there exists a natural \(n\) such that \(x^ n \in Z\) for every \(x\in L\). Let the left \(R\)-module \(L/Z\) be a Noetherian module. Then \(L^ n = L^{n+1} = Z\) for some \(n\in \mathbb{N}\). This result is a partial answer to Herstein's conjecture (1966).