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In this work, we consider an interference-mitigation based cognitive radio system where a secondary transmitter is to communicate with its corresponding receiver without affecting the communication between a primary transmitter-receiver pair. In this case, the secondary transmitter must satisfy a coexistence constraint which requires that no rate degradation occurs at the primary user (PU), even when the latter utilizes only a single-user decoder. To achieve a non-zero rate of the secondary user (SU) under this constraint, Jovicic and Viswanath previously proposed a scheme (referred to as the JV scheme) that utilizes relaying by the secondary transmitter to overcome the interference caused by the simultaneous transmission of the SU's message. In this case, the interference caused by the signals corresponding to PU's message at the secondary receiver (from both the direct and the relay links) is mitigated by employing dirty paper coding (DPC) at the secondary transmitter. However, the interference caused by the SU's message at the primary receiver is not eliminated in this case and, thus, will limit the power (and, hence, the rate) that can be used by the secondary transmitter to transmit its own message. In our work, the use of clean relaying by the secondary transmitter and/or receiver (i.e., relaying without simultaneous transmission of SU's own message) is proposed to improve the quality of the relayed signal and, thereby, increases the rate achievable by the SU. Two improved transmission schemes are proposed: (i) clean relaying by secondary transmitter (CT) and (ii) clean relaying by secondary transmitter and receiver (CTR). The CT scheme utilizes DPC to mitigate interference at the secondary receiver whereas the CTR scheme utilizes coding for multiple access channels with common messages to enable decoding of both PU's and SU's messages at the secondary receiver. The CT scheme can be viewed as a generalization of the JV scheme and, therefore, performs at- least as well as the latter. The CTR scheme, on the other hand, is shown to outperform the CT scheme in terms of the multiplexing gain achievable under full channel state information at the transmitter (CSIT) and in terms of the rate achievable with statistical CSIT. Numerical simulations are provided to illustrate these advantages.