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In this paper, we investigate the problem of spectrally efficient operation of a cognitive radio, also called secondary spectrum user, under an interference from the primary system. A secondary receiver observes a multiple access channel of two users, the secondary and the primary transmitter, respectively. The secondary receiver applies Opportunistic Interference Cancelation (OIC) and Suboptimal Opportunistic Interference Cancelation (S-OIC) thus decoding the primary signal when such an opportunity is created by the rate selected at the primary transmitter and the power received from the primary transmitter. First, we investigate how the secondary transmitter, when using OIC and S-OIC for fixed transmitting power, should select its rate in order to meet its target outage probability under different assumptions about the channel-state-information available at the secondary transmitter. We study three different cases and for each of them identify the region of achievable primary and secondary rates. Second, we determine how the secondary transmitter should select its transmitting power not to violate the target outage probability at the primary terminals. Our numerical results show that the best secondary performance is always obtained when the secondary transmitter knows the instantaneous channel-state-information toward the intended receiver. We also evaluate the degradation in terms of achievable rate at the secondary receiver when it uses suboptimal decoding (S-OIC rather than OIC) and the interplay between the allowed power at the secondary transmitter (which depends on the target outage probability at the primary receiver) and the decodability at the secondary receiver.