In this paper, we investigate rings in which the prime radical is an annihilator and present a characterization of Artinian rings satisfying this property. We also study rings in which the singular ideal and the prime radical coincide. Finally we show that Artinian rings are the direct product of a semiprime ring and a semiprime-free ring (ring in which every nonzero ideal contains a nonzero nilpotent ideal) and present a result on quasi-Baer Artinian rings.