Active reconfigurable intelligent surface (ARIS) has sparked more attention due to its capability to overcome the impact of double fading. This paper introduces using an ARIS to aid non-orthogonal multiple access (NOMA) communications over cascade Rician fading channels, where the direct links between the base station and users are seriously blocked. By applying ARIS, it can amplify the superposed signals to overcome the double pass loss effect caused by passive RIS. Both ARIS and NOMA can have synergistic impacts on sixth-generation communication systems. New approximated and asymptotic expressions in terms of outage probability and ergodic data rate of the k-th user are deduced for ARIS-NOMA systems. Based on asymptotic analytical results, we further calculate the diversity order and high signal-to-noise ratio slope of the k-th user. Finally, the system throughput of ARIS-NOMA is discussed in the delay-constrained transmission mode. Monte Carlo numerical results are performed to verify that: (1) the outage behaviors of ARIS-NOMA are better than that of ARIS-assisted orthogonal multiple access (OMA); (2) as the impact of thermal noise caused by ARIS becomes larger, the communication performance from the base station to ARIS, then to users, becomes worse; (3) the ARIS-NOMA systems have the ability to provide the improved ergodic data rate relative to ARIS-OMA.