The extinction spectra and the electric field distribution of the surface plasmon coupling of gold nanoring dimer in horizontal and vertical arrangements are calculated by the discrete dipole approximation method. It is found that the peaks of extinction spectra and electric field distribution of the surface plasmon coupling are sensitive to the size and the separation gap between gold nanorings. It is demonstrated that the peaks of extinction spectra will be red-shifted or blue-shifted due to the different structure parameters of gold nanorings. Because of the effect of the polarization charge coupling between adjacent gold nanoring, the local electric field distribution is found to be stronger for the gold nanoring dimer and trimer in horizontal arrangement than for the single gold nanoring. The horizontal gold nanoring trimer has stronger enhancement of local electric field than the dimer. It shows that the greater separation gap has the weaker local electric field distribution for the gold nanoring dimer and trimer in horizontal arrangement. Therefore the gold nanoring horizontal array is predicted to be an ideal surface enhanced Raman scattering substrate and is expected to have potential applications in biological and chemical detections.