In this study, we explore the performance of a reconfigurable reflecting surface (RIS)-assisted transmit spatial modulation (SM) system for downlink transmission, wherein the deployment of RIS serves the purpose of blind area coverage within the channel. At the receiving end, we present three detectors, i.e., maximum likelihood (ML) detector, two-stage ML detection, and greedy detector to recover the transmitted signal. By utilizing the ML detector, we initially derive the conditional pair error probability expression for the proposed scheme. Subsequently, we leverage the central limit theorem (CLT) to obtain the probability density function of the combined channel. Following this, the Gaussian-Chebyshev quadrature method is applied to derive a closed-form expression for the unconditional pair error probability and establish the union tight upper bound for the average bit error probability (ABEP). Furthermore, we derive a closed-form expression for the ergodic capacity of the proposed RIS-SM scheme. Monte Carlo simulations are conducted not only to assess the complexity and reliability of the three detection algorithms but also to validate the results obtained through theoretical derivation results.