  We introduce the notion of commuting Ricci tensor for real hypersurfaces in
the complex quadric $Q^m = SO_{m+2}/SO_mSO_2$ . It is shown that the commuting
Ricci tensor gives that the unit normal vector field $N$ becomes $\frak
A$-principal or $\frak A$-isotropic. Then according to each case, we give a
complete classification of real hypersurfaces in $Q^m = SO_{m+2}/SO_mSO_2$ with
commuting Ricci tensor.
