We study a special class of the cognitive radio channel in which the receiver of the cognitive pair does not suffer interference from the primary user. Previously developed general encoding schemes for this channel are complex as they attempt to cope with arbitrary channel conditions, which leads to rate regions that are difficult to evaluate. The focus of our work is to derive simple rate regions that are easily computable, thereby providing more insights into achievable rates and good coding strategies under different channel conditions. We first present several explicit achievable regions for the general discrete memoryless case. We also present an improved outer bound on the capacity region for the case of high interference. We then extend these regions to Gaussian channels. With a simple outer bound we establish a new capacity region in the high-interference regime. Lastly, we provide numerical comparisons between the derived achievable rate regions and the outer bounds.