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We propose an efficient wavelength conversion scheme that is based on either Raman-resonant four-wave mixing or non-resonant Kerr-induced four-wave mixing in a silicon nanowire ring, and that does not require dispersion engineering of the nanowire. We rely on the spatial variation of the Raman and Kerr susceptibilities around the ring to quasi-phase match the wavelength conversion processes for TE polarized fields. The flexibility of this quasi-phase-matching scheme can lead to wavelength conversion efficiencies from -26.7 dB to values larger than 0 dB, and in certain circumstances makes it possible to outperform conventional phase-matched conversion in a dispersion-engineered silicon ring by factors exceeding 6 dB.