  We will consider the torsional completion of gravity for a background filled
with Dirac matter fields, studying what happens when fermionic non-minimal
coupling is taken into account: we will show that, although non-minimal
couplings are usually disregarded because of their ill-defined behaviour in
ultraviolet regimes, this is due to the fact that torsion is commonly
neglected, whereas when torsion is not left aside, even non-minimal couplings
behave properly. In detail, we will see that non-minimal coupling allows to
renormalize the Dirac equation even when torsion is taken into consideration
and that in some type of non-minimally coupled models parity-oddness might be
present even in the gravitational sector. In addition, we will show that in the
presence of the considered non-minimal coupling, torsion is able to evade
cosmological singularities as it can happen in the minimal coupling case and in
some other non--minimally coupled theory. In the course of the paper, we shall
consider a specific interaction as prototype to study this fermionic
non-minimal coupling, but we will try to present results that do not depend on
the actual structure of the non-minimal couplings by investigating alternative
types of interaction.
