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The interactive recommender systems involve users in the recommendation procedure by receiving timely user feedback to update the recommendation policy. Therefore, they are widely used in real application scenarios. Previous interactive recommendation methods primarily focus on learning users' personalized preferences on the relevance properties of an item set. However, the investigation of users' personalized preferences on the diversity properties of an item set is usually ignored. To overcome this problem, we propose the Linear Modular Dispersion Bandit (LMDB) framework, which is an online learning setting for optimizing a combination of modular functions and dispersion functions. Specifically, LMDB employs modular functions to model the relevance properties of each item, and dispersion functions to describe the diversity properties of an item set. Moreover, we also develop a learning algorithm, called Linear Modular Dispersion Hybrid (LMDH) to solve the LMDB problem and derive a gap-free bound on its n-step regret. Extensive experiments on real datasets are performed to demonstrate the effectiveness of the proposed LMDB framework in balancing the recommendation accuracy and diversity.