  It is still an open problem whether the inhibition phenomenon of
Rayleigh--Taylor (RT) instability by horizontal magnetic field can be
mathematically proved in a non-resistive magnetohydrodynamic (MHD) fluid in a
two-dimensional (2D) horizontal slab domain, since it had been roughly verified
by a 2D linearized motion equations in 2012 \cite{WYC}. In this paper, we find
that this inhibition phenomenon can be rigorously verified in the
inhomogeneous, incompressible, inviscid case with velocity damping. More
precisely, there exists a critical number $m_{\rm{C}}$ such that if the
strength $|m|$ of horizontal magnetic field is bigger than $m_{\rm{C}}$, then
the small perturbation solution around the magnetic RT equilibrium state is
exponentially stable in time. Our result is also the first mathematical one
based on the nonlinear motion equations for the proof of inhibition of flow
instabilities by a horizontal magnetic field in a horizontal slab domain. In
addition, we also provide a nonlinear instability result for the case $|m|\in
[0,m_{\rm{C}})$. Our instability result presents that horizontal magnetic field
can not inhibit the RT instability, if it's strength is to small.
