TAILIEUCHUNG - On fuzzy cosets of gamma nearrings
In this paper, we consider fuzzy notion of a Γ-near ring, introduce the notion of a fuzzy coset and obtained some related important fundamental isomorphism theorems. | Turk J Math 29 (2005) , 11 – 22. ¨ ITAK ˙ c TUB On Fuzzy Cosets of Gamma Nearrings Satyanarayana Bhavanari, Syam Prasad Kuncham Abstract In this paper, we consider fuzzy notion of a Γ-near ring, introduce the notion of a fuzzy coset and obtained some related important fundamental isomorphism theorems. Key Words: Gamma nearring, ideal, fuzzy ideal, fuzzy coset, Gamma nearring homomorphism. Introduction A non-empty set N with two binary operations + and · is called a nearring if it satisfies the following axioms. (i) (N, +) is a group (not necessarily Abelian); (ii) (N, ·) is a semi-group; (iii) (a + b)c = ac + bc for all a, b, c ∈ N . Precisely speaking, it is a right near ring. Moreover, a near ring N is said to be a zero-symmetric nearring if n0 = 0 for all n ∈ N , where 0 is the additive identity in N . The concept of Γ-nearring, a generalization of both the concepts nearring and Γ-ring was introduced by Satyanarayana [10]. Later, several authors such as Satyanarayana [11, 12], Booth [1-3] and Booth and Groenewald [4] studied the ideal theory of Γ-nearrings. Let (M, +) be a group (not necessarily Abelian) and Γ a non-empty set. Then M is said to be a Γ-nearring if there exists a mapping M × Γ × M → M (the image of (a, α, b) is denoted by aαb), satisfying the following conditions: 2000 AMS Mathematics Subject Classification: 3E72, 16Y30 11 BHAVANARI, KUNCHAM (i) (a + b)αc = aαc + bαc; (ii) (aαb)βc = aα(bβc) for all a, b, c ∈ M and α, β ∈ Γ. Moreover, M is said to be zero-symmetric if aα0 = 0 for all a ∈ M and α ∈ Γ, where 0 is the additive identity in M. A normal subgroup (I, +) of (M, +) is called (i) a left ideal, if aα(b + i) – aαb ∈ I for all a, b ∈ M, α ∈ Γ, i ∈ I; (ii) a right ideal, if iαa ∈ I for all a ∈ M, α ∈ Γ, i ∈ I; (iii) an ideal, if it is both a left and a right ideal. It is clear that if M is a Γ-nearring, then the elements of Γ act as binary operations on M such that the system (M, +, γ) is a nearring for all γ ∈ Γ. The relations .
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