In CR systems, resource allocation is very important, which can be operated by different rules, according to the different purposes. In this paper, we focus on subcarrier allocation in CR systems, where primary and CR users co-exist in adjacent bands, while keeping the total interference introduced to the PU band below a certain threshold and the total power allocated to the CR users under a constraint. First, we investigate Max-Rate subcarrier allocation algorithm (MaxR-SAA) which can achieve maximum transmit rate of the CR system. Then, focusing on the fact that the less interference introduced to the PU band, the better the PU can work; we proposed a Min-Interference subcarrier allocation algorithm (MinI-SAA). Further, aiming at achieving fairness among all the SUs, we propose a subcarrier allocation algorithm, which is termed as Fair-Rate subcarrier allocation algorithm (FairR-SAA). Numerical results are obtained for the behaviors and performance of our proposed algorithms.