Fuzzy euclidean ideal

ABSTRACTThis work is composed of two chapters; fuzzy subgroups, fuzzy ideals and fuzzyEuclidean ideal.First chapter, which purposes the present basic informations in Literature has eight sections. In the first sectionthe concept of fuzzy subset is given. The relationships between the fuzzy subgroups and level subgroups areanalized in the second section. Then the fuzzy ideals and level ideals are studied. After that fuzzy maximalideal, fuzzy prime ideal and basic theorems about them are given in the fourth and fifth sections. In the sixthsection the concept of fuzzy coset are studied. In the seventh section fuzzy principal ideal is examined. At theend of this chapter the definition of fuzzy relation and fuzzy cartesian product are given. Then the basic theoremsand results about them are studied. Finally some important examples are given.Second chapter, which is the original part of our work, consists of six sections. Firstly,by imposing a new condition on the fuzzy ideal, we define fuzzy Euclidean ideal and givesome examples. After that intersection and union of fuzzy Euclidean ideals arediscussed. In the third section the cartesian products of fuzzy Euclidean ideals areanalized. The relationships between fuzzy prime, fuzzy maksimal, fuzzy quotient ideal,fuzzy principal ideal, fuzzy localized subrings and fuzzy Euclidean ideals are examined.At the end of this chapter the image and pre-image (inverse image) of a fuzzy Euclideanideal under a ring homomorphism f are studied.Keywords: Fuzzy Ideal, Euclidean Ring, Fuzzy Relation, Cartesian Product, Fuzzy Euclidean Ideal.iii