The paper studies decentralized modular control of concurrent discrete event systems that are composed of multiple interacting modules. A modular supervisor consists of a set of local supervisor modules, one for each plant module and which determines its control actions based on the locally observed behaviors. No communication among local supervisor modules occurs in the setting of decentralized modular control. In this paper we introduce the notion of separable-controllability, a property strictly stronger than controllability and separability combined, as a condition for the existence of a decentralized modular control, and present a way to verify this property. We show that non-unique maximal separably-controllable sublanguages and the unique minimal closed and separably-controllable superlanguage of a specification language exist. These serve as an upper bound (resp., the lower bound) for a restrictive (resp., relaxive) decentralized modular control. We present modular computations for synthesizing a restrictive as well as a relaxive decentralized modular control. When appropriate we also compare our results with the existing ones.