The Cognitive Multiple Access Z-Interference Channel consists of a MAC channel that receives interference from an external transmitter. In a previous work we studied this model where both MAC users are cognitive and showed that an encoding scheme based on Dirty Paper coding attains the capacity region for the Gaussian case. In this work we assume that only one MAC user is cognitive and show that the non-cognitive MAC user can still benefit from the cognition of the other MAC user compared to the case where none of the MAC users are cognitive. We compare the achievable rate region obtained for our model with those for the cases where interference is treated as noise and where part of the primary user's message is decodable at the MAC receiver.