Lower and Upper Rough Submodules

In recent several years, the rough set theory has been rapidly developed and applied in pattern recognition, machinery study, decision-making analysis and supporting. The theory of rough sets deals with the approximation of an arbitrary subset of a universe by two definable or observable subsets called lower and upper approximation. In this paper, we shall extend rough sets to module theory by introducing the notions of so called lower and upper rough submodules in a R-module, and give two main results as follows: If A and B are upper rough θ_w-submodules of M, then A+B is an upper rough θ_w-submodule of M; If A and B are lower rough θ_w-submodules of M, then A∩B is an lower rough θ_w-submodule of M.