The increasing focus on carbon neutrality has led to heightened interest in multiple microgrids (MGs) due to their potential to significantly reduce emissions by the integrated electricity-heat-carbon sharing among them. In this paper, a decentralized peer-to-peer (P2P) framework for integrated electricity-heat-carbon sharing is proposed to optimize the trading process of multi-energy and carbon among multiple MGs. The proposed framework considers certified emission reductions (CERs) of photovoltaic (PV) systems in each MG, and carbon allocation and trading among multiple MGs. The P2P trading behaviors among multiple MGs are modelled as a non-cooperative game. A decentralized optimization method is then developed using a price-based incentive scheme to solve the non-cooperative game and optimize the transactions of the electricity-heat-carbon jointly. The optimization problem is solved using sub-gradient in a decentralized manner. And the Nash equilibrium of the non-cooperative game is proven to exist uniquely, ensuring the convergence of the model. Furthermore, the proposed decentralized optimization method safeguards the private information of the MGs. Numerical results show that the total operational cost of the MGs and the carbon emissions can be reduced significantly.