In this paper we consider a real hypersurface $M$ in complex hyperbolic space $H_n mathbb{C}$ 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 $mathrm A$ in $H_n mathbb{C}$ by such a commuting Ricci tensor.

Keywords: real hypersurface, Ricci tensor, Hopf hypersurface