  The seesaw mechanism is the preferred methodology to obtain light neutrino
masses by the introduction of a Majorana mass matrix. Moreover, radiative
corrections can be done in order to improve the predictions of the model.
However, when such a Majorana mass has no inverse, radiative corrections turn
out as the only mechanism to get massive neutrinos. In this letter, it is shown
a nonuniversal abelian extension $\mathrm{U(1)}_{X}$ of the Standard Model in
which neutrinos acquire masses completely by radiative corrections through
one-loop self-energies involving heavier right-handed neutrinos. Moreover, one
right-handed neutrino also gets mass through radiative corrections involving
Majorana fermions. Finally, some limit cases are drafted depending on the
hierarchy among vacuum expectation values and Majorana masses.
