Fuzzy maximal ideals and complete normal fuzzy ideals in Γ-near-rings are considered, and related properties are investigated. . Introduction Γ-near-rings were defined by Satyanarayana, and the ideal theory in Γ-near-rings was studied by Satyanarayana and Booth. | Turk J Math 25 (2001) , 457 – 463. ¨ ITAK ˙ c TUB Fuzzy Maximal Ideals of Gamma Near-Rings∗ ¨ urk Young Bae Jun, Kyung Ho Kim and Mehmet Ali Ozt¨ Abstract Fuzzy maximal ideals and complete normal fuzzy ideals in Γ-near-rings are considered, and related properties are investigated. Key words and phrases: (normal) fuzzy ideal, fuzzy maximal ideal, complete normal fuzzy ideal. 1. Introduction Γ-near-rings were defined by Satyanarayana [16], and the ideal theory in Γ-near-rings was studied by Satyanarayana [16] and Booth [1]. Fuzzy ideals of rings were introduced by Liu [11], and it has been studied by several authors [2, 8, 9, 17]. The notion of fuzzy ideals and its properties were applied to various areas: semigroups [10, 12, 4], BCKalgebras [7, 14], and semirings [5]. In [6], Jun et al. considered the fuzzification of left (resp. right) ideals of Γ-near-rings, and investigated the related properties. Jun et al. [3] also introduced the notion of fuzzy characteristic left (resp. right) ideals and normal fuzzy left (resp. right) ideals of Γ-near-rings, and studied some of their properties. As a continuation of the papers [6] and [3], we state fuzzy maximal ideals and complete normal fuzzy ideals in Γ-near-rings, and investigate its properties. 2. Preliminaries We first recall some basic concepts for the sake of completeness. Recall from [13, p. 3] that a non-empty set R with two binary operations “+”(addition) and “·” (multiplication) is called a near-ring if it satisfies the following axioms: (i) (R, +) is a group, (ii) (R, ·) is a semigroup, 2000 Mathematics Subject Classification: 16Y30, 03E72. paper is dedicated to the memory of Prof. Dr. Mehmet Sapanci. ∗ This 457 ¨ URK ¨ JUN, KIM, OZT (iii) (x + y) · z = x · z + y · z for all x, y, z ∈ R. Precisely speaking, it is a right near-ring because it satisfies the right distributive law. We will use the word “near-ring” to mean “right near-ring”. We denote xy instead of x · y. A Γ-near-ring ([16]) is a triple .
