Fuzzy ideals in gamma rings and derivations in gamma-near-rings

ABSTRACTPh. D. ThesisFUZZY IDEALS IN GAMMA RINGSANDDERIVATIONS IN GAMMA-NEAR-RINGSMustafa UCKUNşË™ouInünü UniversityGraduate School of Natural and Applied SciencesDepartment of Mathematics68 + v pages2005ü üSupervisor: Assist. Prof. Dr. M. Ali OZTURKIn the first chapter of this work, preliminaries are given mainly about the gamma-rings, fuzzy sets, gamma-near-rings and derivations.In the second chapter, various papers which were presented earlier about thefuzzy ideals, symmetric bi-gamma-derivations and near-rings with derivation wassummarized.In the third chapter, characterizations of Artinian and Noetherian gamma-ringsby using fuzzy ideals are given. The notion of fuzzy quotient gamma-rings induced byfuzzy ideals is also introduced, and some results related to gamma-homomorphisms was obtained.In the last chapter, some conditions for a gamma-near-ring with derivation tobe commutative are investigated. Moreover, the notion of a symmetric bi-gamma-derivation (also a symmetric bi-generalized gamma-derivation) of a gamma-near-ringis defined, and using the trace of symmetric bi-gamma-derivations (alsosymmetric bi-generalized gamma-derivation), some results related to a gamma-near-ring to be commutative was obtained.Key Words: (Artinian, Noetherian) gamma-ring, fuzzy ideal, level ideal, fuzzyquotient gamma-ring, gamma-homomorphism, fuzzy point, fuzzy kernel, fuzzyimage, prime gamma-near-ring, gamma-derivation, symmetric bi-gamma-derivation,symmetric bi-generalized gamma-derivation.