Quantum decoherence, which appears when a system interacts with its environment in an irreversible way, plays a fundamental role in the description of quantum-to-classical transitions and has been successfully applied in some important experiments. Here, we study the decoherence in noninertial frames for the first time. It is shown that the decoherence and loss of the entanglement generated by the Unruh effect will influence each other remarkably. It is interesting to note that in the case of the total system under decoherence, the sudden death of entanglement may appear for any acceleration. However, in the case of only Robs qubit underging decoherence sudden death may only occur when the acceleration parameter is greater than a critical point.
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