HUBUNGAN RING ROUGH, SUBRING ROUGH, DAN IDEAL ROUGH DENGAN RING, SUBRING, DAN IDEAL KLASIK

Rough set theory provides a framework for representing uncertainty through approximations of a set based on an equivalence relation. The development of this theory in algebraic structures has led to the concepts of rough rings, rough subrings, and rough ideals; however, the relationships between these three structures and their classical counterparts, particularly those constructed through upper approximations, remain to be clarified. This study aims to examine the relationship between rough algebraic structures and classical algebraic structures based on the construction of upper approximations. The study employs a formal deductive approach by constructing rough algebraic structures from specified sets and equivalence relations and then examining whether the properties of classical rings, subrings, and ideals hold. The results show that the constructed rough algebraic structures do not necessarily satisfy all the properties required of their classical counterparts. Conversely, under certain constructions, rough rings, rough subrings, and rough ideals can simultaneously satisfy the properties of classical rings, subrings, and ideals, respectively. These results identify a conditional relationship between rough and classical algebraic structures, determined by the characteristics of the underlying set and the equivalence relation used to construct the approximations. This study contributes a clearer understanding of the conditions underlying the relationship between rough rings, rough subrings, and rough ideals and their classical counterparts, and provides a basis for further development of algebraic structures based on rough set theory.