Rough ideals under relations associated to fuzzy ideals

In this paper we deal with the theory of rough ideals started in (Davvaz, 2004). We show that the approximation spaces built from an equivalence relation compatible with the ring structure, i.e. associated to a two-sided ideal, are too naive in order to develop practical applications. We propose the use of certain crisp equivalence relations obtained from fuzzy ideals. These relations make available more flexible approximation spaces since they are enriched with a wider class of rough ideals. Furthermore, these are fully compatible with the notion of primeness (semiprimeness). The theory is illustrated by several examples of interest in Engineering and Mathematics.