  In this work, we study the Aharonov-Bohm-Coulomb (ABC) system for a
relativistic Dirac particle with position-dependent mass (PDM). To solve our
system, we use the left-handed and right-handed projection operators. Next, we
explicitly obtain the eigenfunctions and the energy spectrum of the particle.
We verify that these eigenfunctions are written in terms of the generalized
Laguerre polynomials and the energy spectrum depends on the parameters Z,
$\Phi_{AB}$ and $\kappa$. We notice that the parameter $\kappa$ has the
function of increasing the values of the energy levels of the system. In
addition, the usual ABC system is recovered when one considers the limit of
constant mass ($\kappa\to{0}$). Moreover, also we note that even in the absence
of ABC system ($Z=\Phi_{AB}=0$), the particle with PDM still has a discrete
energy spectrum.
