  The most widely used model for granular gases is perhaps the inelastic
hard-sphere model (IHSM), where the grains are assumed to be perfectly smooth
spheres colliding with a constant coefficient of normal restitution. A much
more tractable model is the inelastic Maxwell model (IMM), in which the
velocity-dependent collision rate is replaced by an effective mean-field
constant. This simplification has been taken advantage of by many researchers
to find a number of exact results within the IMM. On the other hand, both the
IHSM and IMM neglect the impact of roughness -- generally present in real
grains -- on the dynamic properties of a granular gas. This is remedied by the
inelastic rough hard-sphere model (IRHSM), where, apart from the coefficient of
normal restitution, a constant coefficient of tangential restitution is
introduced. In parallel to the simplification carried out when going from the
IHSM to the IMM, we propose in this paper an inelastic rough Maxwell model
(IRMM) as a simplification of the IRHSM. The tractability of the proposed model
is illustrated by the exact evaluation of the collisional moments of first and
second degree, and the most relevant ones of third and fourth degree. The
results are applied to the evaluation of the rotational-to-translational
temperature ratio and the velocity cumulants in the homogeneous cooling state.
