Approximations in a module by using set-valued homomorphisms

In applied mathematics, we encounter many examples of mathematical objects that can be added to each other and multiplied by scalar numbers. Modules over a ring conclude all those examples. The initiation and majority of studies on rough sets for algebraic structures such as modules have been concentrated on a congruence relation. The congruence relation, however, seems to restrict the application of the generalized rough set model for algebraic sets. In order to solve this problem, we consider the concept of set-valued homomorphism for modules. The notions of generalized lower and upper approximation operators, constructed by means of a set-valued mapping, which is a generalization of the notion of lower and upper approximations of a module, are provided. We also propose the notion of generalized lower and upper approximations with respect to a submodule of a module and discuss some significant properties of them.