  The exact solutions of the complete (1+3)-dimensional Dirac equation of
fermions moving in ideal Aharonov-Bohm (AB) rings and cylinders are used for
deriving the exact expressions of the relativistic partial currents. It is
shown that these currents can be related to the derivative of the fermion
energy with respect to the flux parameter, just as in the non-relativistic
case. However, a new and remarkable relativistic effect is the saturation of
the partial currents for high values of the total angular momentum. Based on
this property, the total relativistic persistent currents at $T=0$ is evaluated
for rings and cylinders obtaining approximative simple closed formulas.
