  We classify the textures of the Dirac and the right-handed Majorana neutrino
mass matrices, $M_D$ and $M_R$, which can satisfy the so-called ``Strong
Scaling Ansatz'' (SSA) within the framework of the seesaw mechanism
$M_{\nu}=-M_D^T M_R^{-1} M_D$. We assume that the Dirac neutrino mass matrix
has some texture zeros and examine which elements should be zero in order to
satisfy the SSA, by taking into account all possible textures for $M_R$. We
find that the resulting Dirac neutrino mass matrices have rank 2 as well as the
rank of the effective neutrino mass matrix $M_{\nu}$, or rank 1, depending only
on the textures of $M_R^{-1}$. We also consider the three cases of the breaking
of the SSA by introducing a complex breaking parameter in $M_{\nu}$ and show
that it can generate the CP violation in the lepton sector as well as non-zero
$m_3$ and $U_{e3}$. We furthermore discuss the implications of the thermal
leptogenesis for the both case which satisfies and breaks the SSA in the basis
where $M_R$ is diagonal.
