  We derive a general expression for the fermion self-energy in a hot
magnetized plasma by using the Landau-level representation. In the one-loop
approximation, the Dirac structure of the self-energy is characterized by five
different functions that depend on the Landau-level index $n$ and the
longitudinal momentum $p_z$. We derive general expressions for all five
functions and obtain closed-form expressions for their imaginary parts. The
latter receive contributions from three types of on-shell processes, which are
interpreted in terms of Landau-level transitions, accompanied by a single
photon (gluon) emission or absorption. By making use of the imaginary parts of
the self-energy functions, we also derive the Landau-level dependent fermion
damping rates $\Gamma_{n}(p_z)$ and study them numerically in a wide range of
model parameters. We also demonstrate that the two-spin degeneracy of the
Landau levels is lifted by the one-loop self-energy corrections. While the spin
splitting of the damping rates is small, it may be important for some spin and
chiral effects. We argue that the general method and the numerical results for
the rates can have interesting applications in heavy-ion physics, astrophysics,
and cosmology, where strongly magnetized QED or QCD plasmas are ubiquitous.
