With the implementation of the national carbon trading market policy, in order to fully tap the low carbon emission reduction capacity of the regional integrated energy system (RIES) and improve the economic interaction benefits of the multi-regional integrated energy system's access to the active distribution network (ADN), the regional integrated energy system alliance considering flexible resources and low-carbon interaction structure is proposed to participate in the optimal dispatching strategy for distribution network peak shaving. A master-slave game model with active distribution network as the main body and regional integrated energy system alliance as the slave body is established. The main body formulates time-of-use electricity pricing strategies with the goal of maximizing interaction benefits, while the subordinate members achieve multi-energy sharing through interconnection lines. Considering the carbon trading system, the objective is to minimize the combined cost of energy supply and carbon trading, responding to the main body's electricity pricing strategy. A cooperative game optimization model for the multi-regional integrated energy system is established. Flexible resource applications including demand response, energy storage and electric vehicles are introduced, cooperating with carbon capture electric-to-gas coupling units to optimize the low-carbon energy supply strategy of each system within the alliance, and meet the alliance's load demand. Based on Nash bargaining theory, the cooperative benefits among alliance members are distributed. The proposed strategy can effectively reduce the carbon emissions of the regional integrated energy system and ensure the economic benefits of master-slave interaction, as demonstrated through the numerical example.