In this paper we consider a real hypersurface M in complex hyperbolic space $H_nmathbb{C}$ satisfying $S{phi}A;=;{phi}AS$, where $phi$, A and S denote the structure tensor, the shape operator and the Ricci tensor of M respectively. Moreover, we give a characterization of real hypersurfaces of type A in $H_nmathbb{C}$ by such a commuting Ricci tensor.