Amer Himza Almyaly

The backing of the fuzzy ideal is normal ideal in some ring and in same time there fuzzy set whose is not fuzzy ideal and it backing set is ideal, i.e., it crisp is normal ideal. Consequently, in this paper we constructing a fuzziness function which defined on fuzzy sets and assigns membership grade for every fuzzy set whose it backing set are crisp ideal. Now, Let  be collection of all fuzzy subsets of ring  and , and the function  defines from  to, such that the value of the function is greater than zero if the crisp set of fuzzy set is ideal, and the value of the function is equal to one when the crisp set is maximal ideal or is the ring  itself. But if the support of the fuzzy set did not ideal then the value may be equal to or large than 0. Therefore, we add another condition to the fuzziness function to be more determined with respect to the fuzzy set. From above we try to find relation between the fuzzy ideal and its crisp set. This concept is derives from the open grade for all fuzzy set in fuzzy topological space which called smooth topology.