Reconfigurable intelligent surface (RIS) empowered communications with non-orthogonal multiple access (NOMA) has recently become as an appealing research direction for the next-generation wireless communications. In this paper, we propose a novel NOMA solution with RIS partitioning, where we aim to enhance the spectrum efficiency by improving the ergodic rate of all users, and to maximize the user fairness. In the proposed system, we distribute the physical resources among users such that the base station (BS) and RIS are dedicated to serve different clusters of users. Furthermore, we formulate an RIS partitioning optimization problem to slice the RIS elements between the users such that the user fairness is maximized. The formulated problem is a non-convex and non-linear integer programming (NLIP) problem with a combinatorial feasible set, which is very challenging to solve. Therefore, we exploit the structure of the problem to bound its feasible set and obtain a sub-optimal solution by sequentially applying three efficient search algorithms. Furthermore, we derive exact and asymptotic expressions for the outage probability. Simulation results clearly indicate the superiority of the proposed system over the considered benchmark systems in terms of ergodic sum-rate, outage probability, and user fairness performance.