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Within the framework of the complete theory of supersymmetry without R-parity, where all possible R-parity violating terms are admitted, we perform a systematic analytical study of all sources of neutrino masses up to one-loop level. In the passing, we present the full result for squark and slepton masses. In particular, there are interesting LR squark and slepton mixings which involve both bilinear and trilinear R-parity violating parameters and hence have been missing in previous studies under which either one type of the couplings is assume to vanish. The LR mixings play a central role in neutrino mass generations

LR scalar mixings and one-loop neutrino masses