  We investigate the in-plane spin relaxation of electrons due to the
D'yakonov-Perel' and Elliot-Yafet mechanisms including the intra- and
inter-valley processes in monolayer MoS$_2$. We construct the effective
Hamiltonian for the conduction band using the L\"{o}wdin partition method from
the anisotropic two-band Hamiltonian with the intrinsic spin-orbit coupling of
the conduction band included. The spin-orbit coupling of the conduction band
induces the intra- and inter-valley D'yakonov-Perel' spin relaxation. In
addition, the Elliot-Yafet spin relaxation also takes place due to the
interband spin mixing. We find that the D'yakonov-Perel' mechanism dominates
the in-plane spin relaxation. In the framework of this mechanism, the
intravalley process is shown to play a more important role at low temperature
whereas the intervalley one becomes more important at high temperature. At the
temperature in between, the leading process of the in-plane spin relaxation
changes from the intervalley to intravalley one as the electron density
increases. Moreover, we find that the intravalley process is dominated by the
electron-electron Coulomb scattering even with high impurity density since the
dominant term in the spin-orbit coupling is isotropic, which does not lead to
the spin relaxation together with the electron-impurity scattering. This is
very different from the previous studies in semiconductors and graphene.
