Roughness based on fuzzy ideals

The theory of rough set, proposed by Pawlak and the theory of fuzzy set, proposed by Zadeh are complementary generalizations of classical set theory. Many sets are naturally endowed with two binary operations: addition and multiplication. One concept which does this is a ring. This paper concerns a relationship between rough sets, fuzzy sets and ring theory. It is a continuation of ideas presented by Kuroki and Wang [N. Kuroki, P.P. Wang, The lower and upper approximations in a fuzzy group, Inform. Sci. 90 (1996) 203–220]. We consider a ring as a universal set and we assume that the knowledge about objects is restricted by a fuzzy ideal. In fact, we apply the notion of fuzzy ideal of a ring for definitions of the lower and upper approximations in a ring. Some characterizations of the above approximations are made and some examples are presented.