The influence of Rashba spin-orbit interaction (SOI) and polaronic effect on the ground-state energy of electrons in semiconductor quantum rings (QRs) are studied by means of the Lee-Low-Pines variational method. Numerical calculations for GaAs QRs are performed and the results show that the ground-state energy of electrons splits into two branches as E(↑) and E(↓) under the Rashba SOI, which correspond to the spin-up state and spin-down state, respectively. The contribution of the Rashba SOI effect to the ground-state energy of electrons is related to the spin state of electrons and is closely linked to the inner and outer radii of a QR. However, it is independent of the height of the QR. The ground-state energy of electrons decreases due to the polaronic effect in QRs. The energy shift ?Ee-LO of the ground-state of the electron induced by the polaronic effect decreases monotonically with increase of the height of a QR and fluctuates with the changes of the radii of QRs. The amplitude of the fluctuation is very sensitive and remarkable to the changes of the inner radius R1 and the outer radius R2.