  The neutrino mean free path in neutron matter under a strong magnetic field
is evaluated for the inelastic scattering reaction and studied as a function of
the neutron matter density in the range $0.05 \leq \rho \leq 0.4$ fm$^{-3}$ for
several temperatures up to 30 MeV and magnetic field strengths B=0 G, $10^{18}$
G and $2.5\times 10^{18}$ G. Polarized neutron matter is described within the
non--relativistic Brueckner--Hartree--Fock (BHF) approach using the Argonne V18
nucleon-nucleon potential supplemented with the Urbana IX three-nucleon force.
Explicit expressions of the cross section per unit volume for the scattering of
a neutrino with a spin up or spin down neutron are derived from the Fermi
Golden rule. Our results show that the mean free path depends strongly on the
angle of the incoming neutrino, leading to an asymmetry in this quantity. This
asymmetry depends on the magnetic field intensity and on the density, but it is
rather independent of the temperature. For a density of $0.16$ fm$^{-3}$ at a
temperature T$=30$MeV, the asymmetry in the mean free path is found to be of
$\sim 15\%$ for B=$10^{18}$G and $\sim 38\%$ for B=$2.5 \times 10^{18}$G.
