Roughness Results in Rings

A rough set is an approximation of a subset of a universe. Rough sets are mainly used in decision-making when the given data is uncertain. Rough set theory is a groundbreaking approach that provides a formal framework for extracting facts from imperfect data and helps us classify objects based on their similarities. Developing an algebraic structure for rough sets facilitates a detailed study of the settheoretic properties. In this paper, we consider the universe as a ring and obtain rough set results. We consider a new equivalence relation on a ring R whose equivalence classes form a partition of R. Then, we define upper and lower approximations of a subset of a ring R with respect to the given equivalence relation. Subsequently, we prove related results on these approximations and are illustrated with suitable examples. In addition, we obtain the relationship between the upper and lower approximations defined in this paper and the ones defined earlier.