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This letter proposes a low-complexity algorithm for transmit-antenna selection in a multi-input multi-output spatial multiplexing system with per-antenna rate control and an ordered successive interference cancellation receiver. The active antenna subset and its transmission rate are determined at the receiver, and conveyed to the transmitter using limited feedback. We propose a serial antenna-selection procedure consisting of a successive process that tests whether transmit-antenna selection gain exists when the antenna with the lowest signal-to-interference-plus-noise ratio (SINR) is discarded at each selection stage. Furthermore, we show that "reverse-detection ordering", whereby the signal with the lowest SINR is decoded at each stage of successive decoding, widens the disparities among fractions of the whole capacity allocated to each individual antenna, and thus maximizes a gain of antenna selection. Numerical results show that the proposed reverse-detection-ordering-based serial antenna selection induces a negligible capacity loss, compared with the exhaustive selection strategy, even with considerably reduced complexity.