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Lwamo et al. recently proposed a robust and efficient remote single and multi-server biometric authentication scheme using smart card and RSA. The scheme is vulnerable to the smart card lost attacks; therefore, the scheme cannot resist offline guessing attacks and user impersonation attacks, and cannot provide forward security and user anonymity. To address these issues, we propose a new privacy-preserving ring learning with errors (RLWE)-based remote biometric authentication scheme (RRBAS) for single and multi-server environments. RRBAS is the first lattice-based remote biometric authentication scheme for multi-server environments. Security analysis show that RRBAS can satisfy the authenticated key exchange (AKE) security in the random oracle model, resist known security attacks, and provide post-quantum security. The experimental evaluation and comparative analysis show that RRBAS's computational efficiency is better than that of Lwamo et al., while the communication efficiency is slightly lower than traditional schemes because of the large-size ciphertext of the lattice-based cryptosystem, but it is fully capable of session key agreement in single and multi-server environments.