  A relativistic coupled-cluster (RCC) theory is implemented to study electron
impact excitations of atomic species. As a test case, the electron impact
excitations of the $3s ~ ^2S_{1/2} - 3p ~ ^2P_{1/2;3/2}$ resonance transitions
are investigated in the singly charged magnesium (Mg$^+$) ion using this
theory. Accuracies of wave functions of Mg$^+$ are justified by evaluating its
attachment energies of the relevant states and compared with the experimental
values. The continuum wave function of the projectile electron are obtained by
solving Dirac equations assuming distortion potential as static potential of
the ground state of Mg$^+$. Comparison of the calculated electron impact
excitation differential and total cross-sections with the available
measurements are found to be in very good agreements at various incident
electron energies. Further, calculations are carried out in the plasma
environment in the Debye H\"uckel model framework, which could be useful in the
astrophysics. Influence of plasma strength on the cross-sections as well as
linear polarization of the photon emission in the $3p ~ ^2P_{3/2} - 3s ~
^2S_{1/2}$ transition is investigated for different incident electron energies.
