Climate warming nowadays has caused people to increasingly enhance public awareness about carbon emissions from industries. In the storing industry, inventory management for deteriorating items is crucial in the business competition. To slow down the deterioration and ensure the quality of products, the items are usually stored in certain temperature-controlled environment. However, a lot of carbon emissions of the inventory system are caused by these warehousing activities. In a finite panning period, this paper studies a continuous review inventory system and proposes inventory models to analyze the impacts of carbon emissions on inventory system for deteriorating items with a general time-varying demand, in which shortages are allowed, and the customer demand during shortage period is partially backlogged till the next replenishment. Under carbon emission regulations, the existence and uniqueness of the optimal solution to each model is explored and comparisons of optimality among the proposed models are given. Numerical examples and robust analysis of the models are presented to illustrate the applicability in practice. Some management insights about inventory policies and emission reduction strategies are obtained.