I. N. Herstein conjectured [in J. Math. Anal. Appl. 15, 91-96 (1966; Zbl 0139.260)] that if \(I\subset J\) are left ideals of a left Noetherian ring with some power of each element of J lying in I, then \(J^ n\subseteq I\) for some n; he also proved the conjecture when R satisfies a polynomial identity. In the present note some very pretty arguments are brought to bear to prove Herstein’s conjecture when J/I is an Artinian left R-module. As corollaries, the conjecture is verified when R is left fully bounded, or simple, or the integral group ring of a polycyclic-by-finite group.