  Shear flow instabilities can profoundly affect the diffusion of momentum in
jets, stars, and disks. The Richardson criterion gives a sufficient condition
for instability of a shear flow in a stratified medium. The velocity gradient
$V'$ can only destabilize a stably stratified medium with squared Brunt-Vaisala
frequency $N^2$ if $V'^2/4>N^2$. We find this is no longer true when the medium
is a magnetized plasma. We investigate the effect of stable stratification on
magnetic field and velocity profiles unstable to magneto-shear instabilities,
i.e., instabilities which require the presence of both magnetic field and shear
flow. We show that a family of profiles originally studied by Tatsuno & Dorland
(2006) remain unstable even when $V'^2/4<N^2$, violating the Richardson
criterion. However, not all magnetic fields can result in a violation of the
Richardson criterion. We consider a class of flows originally considered by
Kent (1968), which are destabilized by a constant magnetic field, and show that
they become stable when $V'^2/4<N^2$, as predicted by the Richardson criterion.
This suggests that magnetic free energy is required to violate the Richardson
criterion. This work implies that the Richardson criterion cannot be used when
evaluating the ideal stability of a sheared, stably stratified, and magnetized
plasma. We briefly discuss the implications for astrophysical systems.
