  It is still open whether the phenomenon of inhibition of Rayleigh--Taylor
(RT) instability by a horizontal magnetic field can be mathematically verified
for a non-resistive \emph{viscous} magnetohydrodynamic (MHD) fluid in a
two-dimensional (2D) horizontal slab domain, since it was roughly proved in the
linearized case by Wang in \cite{WYC}. In this paper, we prove such inhibition
phenomenon by the (nonlinear) inhomogeneous, incompressible, \emph{viscous
case} with \emph{Navier (slip) boundary condition}. More precisely, we show
that there is a critical number of field strength $m_{\mm{C}}$, such that if
the strength $|m|$ of a horizontal magnetic field is bigger than $m_{\mm{C}}$,
then the small perturbation solution around the magnetic RT equilibrium state
is {algebraically} stable in time. In addition, we also provide a nonlinear
instability result for the case $|m|\in[0, m_{\mm{C}})$. The instability result
presents that a horizontal magnetic field can not inhibit the RT instability,
if it's strength is too small.
