In this paper, we consider the precoder design for an under-loaded or critically loaded downlink multi-user multiple-input multiple-output (MU-MIMO) communication system. We propose novel precoding and decoding schemes which enhance system performance based on rate splitting at the transmitter and single-stage successive interference cancellation at the receivers. The proposed successive null-space (SNS) precoding scheme utilizes linear combinations of the null-space basis vectors of the successively augmented MIMO channel matrices of the users as precoding vectors to adjust the inter-user-interference experienced by the receivers. We formulate a non-convex weighted sum rate (WSR) optimization problem, and solve it via successive convex approximation to obtain a suboptimal solution for the precoding vectors and the associated power allocation. Our simulation results reveal that the proposed SNS precoders outperform block diagonalization based linear and rate splitting designs, and in many cases, have a relatively small gap to the maximum sum rate achieved by dirty paper coding.