Influences of Rashba spin-orbit interaction(SOI) effects on the ground state energy of the electron in quantum ring at zero fields are studied by means of the perturbation method. Numerical calculations for CdF2 ring are performed and the results show that the Rashba SOI makes the ground state energy of electron split into two branches, which are induced by the spin-up state and spin-down state, respectively. Two branches splitting energy increases oscillatorily with increasing the inner radius of quantum ring and decreases oscillatorily with increasing the outer radius of quantum ring. The largest Rashba spin splitting energies in the process of oscillation-increase and oscillation-decrease are 11.91meV and 13.97meV, respectively. These data and results are helpful to design and develop spin field effect transistor, spin light-emitting diode, spin resonant tunneling device, etc.