ON FUZZY k−IDEALS, k−FUZZY IDEALS AND FUZZY 2−PRIME IDEALS IN Γ−SEMIRINGS

Abstract

The notion of Γ-semiring was introduced by M. Murali Krishna Rao [8] as a generalization of Γ-ring as well as of semiring. In this paper fuzzy k-ideals, k-fuzzy ideals and fuzzy-2-prime ideals in Γ-semirings have been introduced and study the properties related to them. Let μ be a fuzzy k-ideal of Γ-semiring M with |Im(μ)| = 2 and μ(0) = 1. Then we establish that M_μ is a 2-prime ideal of Γ-semiring M if and only if μ is a fuzzy prime ideal of Γ-semiring M.

1. Introduction

Semiring is one of the universal algebras which is a generalization not only of ring but also of distributive lattice. As an universal algebra (S,+, ·) is called a semiring if and only if (S, +), (S, ·) are semigroups which are connected by distributive laws, i.e., a(b + c) = ab + ac, (a + b)c = ac + bc, for all a, b, c ∈ S. Semiring was first introduced by H. S. Vandiver [14] in 1934. Though semiring is a generalization of ring, ideals of semiring do not coincide with ring ideals. For example an ideal of semiring need not be the kernel of some semiring homomorphism. To solve this problem Henriksen [4] defined k−ideals and Iijuka [5] defined h−ideals in semirings to obtain analogues of ring results for semirings. The theory of rings and the theory of semigroups have considerable impact on the development of the theory of semirings. Semiring is very useful for solving problems in applied mathematics and information science because semiring provides an algebraic frame work for modeling. Semirings play an important role in studying matrices and determinants.

The notion of Γ-ring was introduced by Nobusawa [10] as a generalization of ring in 1964. Sen [12] introduced the notion of Γ- semigroup in 1981. The notion of ternary algebraic system was introduced by Lehmer [6] in 1932, Lister [7] introduced ternary ring. Dutta & Kar [3] introduced the notion of ternary semiring which is a generalization of ternary ring and semiring. In 1995, Murali Krishna Rao [8] introduced the notion of Γ- semiring which is a generalization of Γ- ring, ternary semiring and semiring.

The theory of fuzzy sets is the most appropriate theory for dealing with uncertainty was first introduced by Zadeh [15]. The concept of fuzzy subgroup was introduced by Rosenfeld [11]. Many papers on fuzzy sets appeared showing the importance of the concept and its applications to logic, set theory, group theory, ring theory, real analysis, topology, measure theory etc . Uncertain data in many important applications in the areas such as economics, engineering, environment, medical sciences and business management could be caused by data randomness, information incompleteness, limitations of measuring instrument, delayed data updates etc. Swamy and Swamy [13] studied fuzzy prime ideal of rings. Dheena et al. [2] studied fuzzy 2−prime ideal in semirings. Murali krishna Rao [9] studied fuzzy soft Γ-semirings and fuzzy soft k−ideals over Γ-semirings. In this paper fuzzy k−ideals, k−fuzzy ideals and fuzzy-2−prime ideals in Γ-semirings have been introduced and study the properties related to them. We study the homomorphic images and pre-images of fuzzy k−ideals and k−fuzzy ideals of Γ-semirings. Let μ be a fuzzy k−ideal of Γ-semiring M with |Im(μ)| = 2 and μ(0) = 1. Then we establish that Mμ is a 2−prime ideal of Γ-semiring M if and only if μ is a fuzzy prime ideal of Γ-semiring M.

 

2. Preliminary Results

In this section we will recall some of the fundamental concepts and definitions, these are necessary for this paper.

Definition 2.1 ([1]). A set R together with two associative binary operations called addition and multiplication (denoted by + and · respectively) will be called a semiring provided

Definition 2.2 ([8]). Let (M, +) and (Γ, +) be commutative semigroups. Then we call M as a Γ-semiring, if there exists a mapping M×Γ×M → M written as (x, α, y) as xαy such that it satisfies the following axioms for all x, y, z ∈ M and α, β ∈ Γ.

Every semiring R is a Γ-semiring with Γ = R and ternary operation xγy as the usual semiring multiplication. A Γ-semiring M is said to have zero element if there exists element 0 ∈ M such that 0 + x = x = x + 0 and 0αx = xα0 = 0 for all x ∈ M. A Γ-semiring M is said to be commutative Γ-semiring if xαy = yαx for all x, y ∈ M and α ∈ Γ. An additive subsemigroup I of Γ-semiring M is said to be a left (right) ideal of Γ-semiring M if MΓI ⊆ I (IΓM ⊆ I). If I is both left and right ideal then I is called an ideal of Γ-semiring M. An ideal I of Γ-semiring M is called a k−ideal, if b ∈ M, a+b ∈ I and a ∈ I then b ∈ I. A function f : R → S where R and S are Γ-semirings is said to be a Γ-semiring homomorphism if f(a + b) = f(a) + f(b) and f(aαb) = f(a)αf(b) for all a, b ∈ R, α ∈ Γ.

Let S be a nonempty set. A mapping f : S → [0, 1] is called a fuzzy subset of S. Let A be a subset of S. The characteristic function χA of A is a fuzzy subset of S is defined by Let f be a fuzzy subset of a nonempty set S for t ∈ [0, 1], the set ft = {x ∈ S | f(x) ≥ t} is called level subset of S with respect to f. For any x ∈ M and t ∈ [0, 1], we define the fuzzy point xt as If xt is a fuzzy point and μ is any fuzzy subset of Γ-semiring M and xt ⊆ μ then we write xt ∈ μ. xt ∈ μ if and only if x ∈ μt. Let μ be a fuzzy subset of R. Then the image of μ denoted by Im(μ) = {μ(r) | r ∈ R} and |Im(μ)| denotes the cardinality of Im(μ). A fuzzy subset μ : S → [0, 1] is a nonempty, if μ is not the constant function. For any two fuzzy subsets λ and μ of S, λ ⊆ μ means λ(a) ≤ μ(a) for all a ∈ S. Let S and R be two nonempty sets and ψ : S → R be any function. A fuzzy subset f of S is called a ψ invariant if ψ(x) = ψ(y) ⇒ f(x) = f(y) for all x, y ∈ S. Let f and g be two fuzzy subsets of Γ-semiring M. the product fg is defined by ; for x, y, z ∈ M, α ∈ Γ.

 

3. MAIN RESULTS

In this section fuzzy-2−prime ideals, fuzzy k−ideals and k−fuzzy ideals in Γ-semirings have been introduced and study the properties related to them. Throughout this paper M is a commutative Γ-semiring with zero element.

Definition 3.1. A fuzzy subset f of Γ-semiring M is called a fuzzy ideal of Γ-semiring M, if for all x, y ∈ M, α ∈ Γ

Definition 3.2. A fuzzy ideal f of Γ-semiring M is said to be fuzzy k−ideal of Γ-semiring M, if f(x) ≥ min{f(x + y), f(y)} for all x, y ∈ M.

This definition can also be written as a fuzzy ideal f of Γ-semiring M is said to be fuzzy k−ideal of Γ-semiring M if f(x + y) ≥ λ, f(y) ≥ λ ⇒ f(x) ≥ λ for all x, y ∈ M, λ ∈ [0, 1].

Definition 3.3. A fuzzy ideal f of Γ-semiring M is said to be k−fuzzy ideal of Γ-semiring M if f(x + y) = f(0) and f(y) = f(0) ⇒ f(x) = f(0) for all x, y ∈ M.

Example 3.4. Let M be the additive commutative semigroup of all non negative integers and Γ be the additive commutative semigroup of all natural numbers. Ternary operation is defined by aαb = product of a, α, b, for all a, b ∈ M and α ∈ Γ. Then M is a Γ-semiring.

Let μ be a fuzzy subset of Γ-semiring M, defined by

Then μ is a fuzzy k−ideal of Γ-semiring M and μ is also k−fuzzy ideal of Γ-semiring M. λ is a fuzzy ideal but not a fuzzy k−ideal.

Definition 3.5. The ideal generated by a, a ∈ M is defined as the smallest ideal of Γ-semiring M which contains a and it is denoted by (a). The k−ideal generated by a, a ∈ M is defined as the smallest k−ideal of Γ-semiring M which contains a and it is denoted by (a)k

Definition 3.6. If A is an ideal of Γ-semiring M then for some x ∈ A} is called a k−closure of A.

Definition 3.7. Let M be a Γ-semiring and f be a fuzzy ideal of Γ-semiring M. The k−fuzzy closure of f is defined by

Definition 3.8. An ideal P of Γ-semiring M is called a prime ideal if for any ideals A,B of Γ-semiring M, AΓB ⊆ P then A ⊆ P or B ⊆ P

Definition 3.9. An ideal P of Γ-semiring M is called a 2−prime ideal if for any k−ideals A,B of Γ-semiring M, AΓB ⊆ P then A ⊆ P or B ⊆ P

Definition 3.10. A fuzzy ideal μ of Γ-semiring M is called a fuzzy prime ideal if for any fuzzy ideals f, g of Γ-semiring M, fg ⊆ μ then f ⊆ μ, g ⊆ μ.

Definition 3.11. A fuzzy ideal μ of Γ-semiring M is called a fuzzy 2−prime ideal if for any fuzzy k−ideals f, g of Γ-semiring M, fg ⊆ μ then f ⊆ μ or g ⊆ μ.

Definition 3.12. Let M and N be Γ-semirings, ϕ : M → N be a homomorphism of Γ-semiring M and f be a subset of Γ-semiring M. We define a fuzzy subset ϕ(f) of Γ-semiring N by

We call ϕ(f) is the image of f under ϕ.

Definition 3.13. Let ϕ : M → N be a homomorphism of Γ-semiring and f be a fuzzy subset of Γ-semiring N. We define a fuzzy subset ϕ−1(f) of Γ-semiring M by ϕ−1(f)(x) = f(ϕ(x)) for all x ∈ M, we call ϕ−1(f) is a pre-image of f.

We now state the following lemmas, proofs are which are analogous to the corresponding lemmas in semirings [1] similar, so we omit the proofs.

Lemma 3.14. If A is an ideal of Γ-semiring M then is a k−ideal of Γ-semiring M.

Lemma 3.15. If A is an ideal of Γ-semiring M then A is a k−ideal if and only if

Lemma 3.16. Let f1 and f2 be any two fuzzy subsets of Γ-semiring M. If f1 ⊆ f, f2 ⊆ g then f1f2 ⊆ fg for any fuzzy subsets f and g of Γ-semiring M.

Lemma 3.17. If μ is a fuzzy prime ideal of Γ-semiring M then μ is a fuzzy 2−prime ideal of Γ-semiring M.

Lemma 3.18. If μ is a fuzzy ideal of Γ-semiring M and a ∈ M then μ(x) > μ(a) for all x ∈ (a).

Lemma 3.19. Let f, g be any fuzzy ideals of Γ-semiring M and μ be a fuzzy k−ideal of Γ-semiring M. If fg ⊆ μ then

Lemma 3.20. Let μ be a fuzzy subset of Γ-semiring M. Then μ is a fuzzy prime ideal if and only if Mμ = {x ∈ M | μ(x) = μ(0)} is a prime ideal of Γ-semiring M.

Lemma 3.21. Let I be an ideal of Γ-semiring M and α < β ≠ 0. If the fuzzy subset μ of Γ-semiring M is defined by Then μ is a fuzzy ideal of Γ-semiring M.

Theorem 3.22. Let f be a fuzzy ideal of Γ-semiring M. Then f(x) ≤ f(0) for all x ∈ M.

Proof. Let x ∈ M, α ∈ Γ, f(0) = f(0αx) ≥ f(x). Therefore f(x) ≤ f(0), for all x ∈ M. □

Theorem 3.23. Let f and g be fuzzy ideals of Γ-semiring M. If f and g are fuzzy ideals of Γ-semiring M then f ∩ g is a fuzzy ideal of Γ-semiring M.

Proof. Let f and g be fuzzy ideals of Γ-semiring M, x, y ∈ M and α ∈ Γ. Then

Hence f ∩ g is a fuzzy ideal of Γ-semiring M. □

Theorem 3.24. Let f and g be fuzzy k−ideals of Γ-semiring M. Then f ∩ g is a fuzzy k−ideal of Γ-semiring M.

Proof. Let f and g are fuzzy k−ideals of Γ-semiring M. By Theorem [3.23], f ∩ g is a fuzzy ideal of Γ-semiring M. Let x, y ∈ M. We have

Hence f ∩ g is a fuzzy k−ideal of Γ-semiring M. □

Theorem 3.25. Let μ be a fuzzy k−ideal of Γ-semiring M and a ∈ M. Then μ(x) > μ(a) for all x ∈ (a)k.

Proof. Let μ be a fuzzy k−ideal of Γ-semiring M and a ∈ M. If x ∈ (a)k then x + y ∈ (a) for some y ∈ (a), by Lemma 3.18, μ(x + y) > μ(a) and μ(y) > μ(a).

Hence the theorem. □

Theorem 3.26. Let μ be a fuzzy ideal of Γ-semiring M. If xryt ⊆ μ ⇒ xr ⊆ μ or yt ⊆ μ then μ is a fuzzy prime ideal of Γ-semiring M.

Proof. Let σ and θ be fuzzy ideals of Γ-semiring M and σθ ⊆ μ.

Suppose σ ⊈ μ. Then there exists x ∈ M ∋ σ(x) ≥ μ(x).

Let σ(x) = a, y ∈ M and θ(y) = b. If z = xγy, for some γ ∈ Γ then (xayb)(z) = min{a, b}. Therefore

Therefore

Hence μ is a fuzzy prime ideal of Γ-semiring M. □

Theorem 3.27. Let I be an ideal of Γ-semiring M, α ∈ [0, 1). If μ be a fuzzy subset of Γ-semiring M defined by If I is a 2−prime ideal then μ is a fuzzy 2−prime ideal of Γ-semiring M.

Proof. Let I be a 2−prime ideal of Γ-semiring M. Clearly μ is a non constant fuzzy ideal of Γ-semiring M. Let σ, θ be fuzzy ideals of Γ-semiring M such that σθ ⊆ μ, σ ⊈ μ and θ ⊈ μ.

Then there exist x, y ∈ M such that

since I is a 2−prime ideal of Γ-semiring M, γ ∈ Γ. Therefore there exist c ∈ (x)k, d ∈ (y)k such that cγd ∉ I.

Let a = cγd, μ(a) = μ(cγd) = α, σθ(a) ≤ μ(a) = α. Now

Which is a contradiction to the fact that σθ ⊆ μ.

Hence μ is a fuzzy 2−prime ideal of Γ-semiring M. □

Corollary 3.28. Let I be an ideal of Γ-semiring M. If I is a 2−prime ideal of Γ-semiring M then the characteristic function χ1 is a fuzzy 2−prime ideal of Γ-semiring M.

Lemma 3.29. If I is an ideal of Γ-semiring M then χI is a fuzzy ideal of Γ-semiring M.

Proof. Suppose I is an ideal of Γ-semiring M and a, b ∈ I. Then a+b ∈ I, aαb ∈ I for all α ∈ Γ.

Suppose a ∈ I, b ∉ I, α ∈ Γ and a + b ∉ I. Then

Hence χI is a fuzzy ideal of Γ-semiring M. □

Lemma 3.30. If t ∈ [0, 1] such that ft ≠ ϕ, ft is a k−ideal of Γ-semiring M then f is a k−fuzzy ideal of Γ-semiring M.

Proof. Let M be a Γ-semiring, f(x + y) = f(0) and f(y) = f(0). Then

We have f(x) ≤ f(0). Therefore f(x) = f(0). Hence f is a k−fuzzy ideal of Γ-semiring M. □

Lemma 3.31. Let w be a fixed element of Γ-semiring M. If μ is a fuzzy k−ideal of Γ-semiring M then μw = {x ∈ μ | μ(x) ≥ μ(w)} is a k−ideal of Γ-semiring M.

Proof. Let w be a fixed element of Γ-semiring M. and x, y ∈ μw. Then

Let x ∈ μw, r ∈ M and α ∈ Γ. Then

Therefore μw is an ideal of M.

Let x, x + y ∈ μw. Then

Therefore y ∈ μw. Hence μw is a k−ideal of Γ-semiring M. □

Theorem 3.32. Let M be a Γ-semiring and I ⊆ M. Then I is a k−ideal of Γ-semiring M if and only if χI is a k−fuzzy ideal of Γ-semiring M.

Proof. Let I be a k−ideal of Γ-semiring M. Then χI is a fuzzy ideal by Lemma 3.29. Let x, y ∈ M.

Now χI (y) = χI(0) = 1 ⇒ y ∈ I.

Now x + y ∈ I and y ∈ I. Therefore x ∈ I, since I is a k−ideal. Then χI (x) = 1 = χI(0) and hence χI is a k−fuzzy ideal of Γ-semiring M.

Conversely, suppose that χI is a k−fuzzy ideal of Γ-semiring M ⇒ I is an ideal of Γ-semiring M ⇒ χI(0) = 1, since 0 ∈ I. Suppose x + y and y ∈ I. Then χI (x + y) = χI (y) = χI(0) ⇒ χI (x) = 0, since χI is a k−fuzzy ideal of Γ-semiring M. Therefore x ∈ I. Hence I is a k−ideal of Γ-semiring M. □

Theorem 3.33. f is a fuzzy ideal of Γ-semiring M if and only if for any t ∈ [0, 1] such that ft ≠ φ, ft is an ideal of Γ-semiring M.

Proof. Let ft be an ideal of Γ-semiring M, x, y ∈ M and t = min{f(x), f(y)}. Then

Let s = max{f(x), f(y)}, α ∈ Γ. Then

Hence f is a fuzzy ideal of Γ-semiring M.

Conversely suppose that f is a fuzzy ideal of Γ-semiring M and t ∈ [0, 1] so that

Let x ∈ ft, y ∈ M | ft then f(x) ≥ t, α ∈ Γ. Then

Hence ft is an ideal of Γ-semiring M. □

Theorem 3.34. A fuzzy subset μ of Γ-semiring M is a fuzzy k−ideal of Γ-semiring M if and only if μt is a k−ideal of Γ-semiring M for any t ∈ [0, 1], μt ≠ ϕ.

Proof. Let μ be a fuzzy k−ideal of Γ-semiring M. Clearly μt is an ideal of Γ-semiring M, by Theorem 3.33. Suppose a, a + x ∈ μt, x ∈ M ⇒ μ(a) ≥ t, μ(a + x) ≥ t. Since μ is a fuzzy k−ideal, we have

Hence μt is a k−ideal of Γ-semiring M.

Conversely, assume that μt is a k−ideal of Γ-semiring M for any t ∈ [0, 1] with μt ≠ ϕ. Let μ(a) = t1, μ(x + a) = t2 and t = min{t1, t2}.

Then a ∈ μt and a + x ∈ μt for some x ∈ M, since μt is a k−ideal, we have x ∈ μt, μ(x) ≥ min{μ(x+a), μ(a)}. Therefore μ is a fuzzy k−ideal of Γ-semiring M. □

Theorem 3.35. Let μ be a fuzzy k−ideal of Γ-semiring M. Then two k−ideals μs, μt of Γ-semiring M (with s < t, s, t ∈ [0, 1]) are equal if and only if there is no x ∈ M such that s ≤ μ(x) < t.

Proof. Let μ be a fuzzy k−ideal of Γ-semiring M and two k−ideals μs, μt of Γ-semiring M. Suppose s < t ∈ [0, 1] and μs = μt. If there exists x ∈ M such that s < μ(x) < t. Then μt is a proper subset of μs, which is a contradiction. Therefore they are equal.

Conversely, suppose that there is no x ∈ M such that s < μ(x) < t. Then we have s < t ⇒ μs ⊂ μt. If x ∈ μs then μ(x) ≥ s and no μ(x) ≥ t. Since μ(x) ≮ t,⇒ x ∈ μt. Hence μs = μt. □

Theorem 3.36. Let μ be a fuzzy k−ideal of Γ-semiring M. If Im(μ) = {t1, t2, · · · , tn} where t1 < t2 < t3 · · · < tn then family of k−ideals μti , i = 1, 2, 3, · · · , n, is a collection of all level ideals of Γ-semiring M.

Proof. If t ∈ [0, 1] with t < t1 then μt1 ⊂ μt. Since μt1 = M, μt = M. If t ∈ [0, 1] with ti < t < ti+1 then there is no x ∈ M such that ti < μ(x) < ti+1. It follows μti ⊆ μti+2. Hence family of k-ideals μti , i = 1, 2, · · · , n, is collection of all level ideals of Γ-semiring M. □

Theorem 3.37. Let M be a Γ-semiring, t ∈ [0, 1] and ft be a k−ideal of Γ-semiring M. Then f is a k−fuzzy ideal of Γ-semiring M.

Proof. Let ft be a k−ideal of Γ-semiring M, x, y ∈ M. Suppose f(x + y) = f(0), f(y) = f(0) and t = f(0). Then x + y ∈ ft, y ∈ ft.

Since ft is a k−ideal of Γ-semiring M, we have x ∈ ft ⇒ f(x) ≥ t = f(0), we have f(x) ≤ f(0) for all x ∈ M. Therefore f(x) = f(0). Hence ft is a k−fuzzy ideal of Γ-semiring M. □

Theorem 3.38. If I be a k−ideal of Γ-semiring M then there exists a fuzzy k−ideal μ of Γ-semiring M such that μt = I for some t ∈ (0, 1].

Proof. We define a fuzzy subset μ of Γ-semiring M by

Clearly μt = I. If s ∈ (0, 1] then μs is a k−ideal of Γ-semiring M. Hence, by Theorem 3.34, fuzzy subset μ is a fuzzy k−ideal. □

Theorem 3.39. Let M and N be Γ-semirings, ϕ : M → N be a homomorphism and f be a ϕ invariant fuzzy ideal of Γ-semiring M. If x = ϕ(a) then ϕ(f)(x) = f(a), a ∈ M.

Proof. Let M and N be Γ-semirings, a ∈ M, x ∈ N, x = ϕ(a). Then a ∈ ϕ−1(x) and t ∈ ϕ−1(x). Therefore ϕ(t) = x = ϕ(a), since f is ϕ invariant.

Hence ϕ(f)(x) = f(a). □

Theorem 3.40. Let M and N be Γ-semirings and ϕ : M → N be an onto homomorphism. If f is a ϕ invariant fuzzy ideal of Γ-semiring M then ϕ(f) is a fuzzy ideal of Γ-semiring N.

Proof. Let M and N be Γ-semirings, ϕ : M → N be an onto homomorphism and x, y ∈ N. Then there exist a, b ∈ M such that ϕ(a) = x, ϕ(b) = y.

Hence ϕ(f) is a fuzzy ideal of Γ-semiring N. □

Theorem 3.41. Let M and N be Γ-semirings, ϕ : M → N be an onto homomorphism and f be a ϕ invariant fuzzy ideal of Γ-semiring M. Then f is a k−fuzzy ideal if and only if ϕ(f) is a k−fuzzy ideal of Γ-semiring N.

Proof. Let M and N be Γ-semirings, ϕ : M → N be an onto homomorphism and f be a ϕ invariant fuzzy ideal of Γ-semiring M. By Theorem 3.40, ϕ(f) is a fuzzy ideal of Γ-semiring N. Let x, y ∈ N. Then there exist a, b ∈ M such that ϕ(a) = x, ϕ(b) = y.

Since ϕ(a + b) = ϕ(a) + ϕ(b) = x + y,

Since f is a k−ideal of Γ-semiring M,

Suppose ϕ(f) is a k−fuzzy ideal of Γ-semiring N. Then ϕ−1(ϕ(f)) is a k−fuzzy ideal of Γ-semiring M. Therefore f is a k−fuzzy ideal of Γ-semiring M. □

Theorem 3.42. Let ϕ : M → N be an onto homomorphism of Γ-semirings and f be the fuzzy k−ideal of of Γ-semiring N. Then ϕ−1(f) is a fuzzy k−ideal of Γ-semiring M.

Proof. Let ϕ : M → N be an onto homomorphism of Γ-semirings and f be the fuzzy k−ideal of of Γ-semiring N. By Definition 3.13, ϕ−1f(x+y) = f{ϕ(x+y)} for all x, y ∈ M

Let a, b ∈ N. Then there exist x, y ∈ M such that ϕ(x) = a, ϕ(y) = b.

Hence ϕ−1(f) is a fuzzy k−ideal of Γ-semiring M. □

Theorem 3.43. Let ϕ : M → N be an onto homomorphism of Γ-semiring and f be the fuzzy ideal of Γ-semiring N. If ϕ−1(f) is a k−fuzzy ideal of Γ-semiring M then f is a k−fuzzy ideal of Γ-semiring N.

Proof. Let ϕ : M → N be an onto homomorphism of Γ-semiring, f be the fuzzy ideal of N and ϕ−1(f) be a k−fuzzy ideal of Γ-semiring M. Suppose x, y ∈ N, f(x+y) = f(0) and f(y) = f(0). Since ϕ is onto, there exist a, b ∈ M such that ϕ(a) = x, ϕ(b) = y,

Therefore f is a k−fuzzy ideal of Γ-semiring N. □

Theorem 3.44. Let M be a Γ-semiring . Then f is a k−fuzzy ideal of Γ-semiring M if and only if

Proof. Let M be a Γ-semiring, f be a k−fuzzy ideal of M and Then since then there exist a, b ∈ ff(0) such that a + x = b ⇒ f(a) = f(0), f(a+x) = f(b) = f(0) then f(x) = f(0). Therefore

Conversely suppose that Then ff(0) is a k−ideal of Γ-semiring M. Let x, y ∈ M. Then

Hence f is a k−fuzzy ideal of Γ-semiring M. □

Let M be a Γ-semiring and μ be a fuzzy subset of Γ-semiring M. The set {x | μ(x) = μ(0)} is denoted by Mμ.

Theorem 3.45. Let μ be a fuzzy subset of Γ-semiring M, |Im(μ)| = 2 and μ(0) = 1. If Mμ is an ideal of Γ-semiring M then μ is a fuzzy k−ideal of Γ-semiring M.

Proof. Let μ be a fuzzy subset of Γ-semiring M, |Im(μ)| = 2, μ(0) = 1, Im(μ) = {t, 1}, t < 1 and x, y ∈ M. If x, y ∈ Mμ then

Let x ∈ Mμ, y ∉ Mμ and x+y ∈ Mμ. Then μ(x+y) ≥ min{μ(x), μ(y)} = μ(0). Suppose x + y ∉ Mμ. Then μ(x + y) ≥ min{μ(x), μ(y)}. If x ∉ Mμ and y ∉ Mμ then μ(x + y) ≥ min{μ(x), μ(y)}. Hence μ(x + y) ≥ min{μ(x), μ(y)} for all x, y ∈ M.

Let x ∈ Mμ, y ∈ M, α ∈ Γ. Then μ(x) = μ(0), μ(xαy) ≥ max{μ(x), μ(y)}. Let x, y ∈ M, x ∉ Mμ and y ∉ Mμ. Then

Let x, y ∈ M, x ∈ Mμ and y ∈ Mμ. Then

Let x, y ∈ M, x ∉ Mμ and y ∈ Mμ. Then x + y ∉ Mμ, μ(x) ≥ t. Hence μ is a fuzzy k−ideal of Γ-semiring M. □

Theorem 3.46. Let μ be a fuzzy k−ideal of Γ-semiring M. Then Mμ is a k−ideal of Γ-semiring M.

Proof. Let μ be a fuzzy k−ideal of Γ-semiring M, x, y ∈ Mμ and α ∈ Γ. Then

We have μ(0) ≥ μ(x + y). Therefore μ(0) = μ(x + y) ⇒ x + y ∈ Mμ. μ(xαy) ≥ max{μ(x), μ(y)} = max{μ(0), μ(y)} = μ(0), we have

Therefore Mμ is an ideal of Γ-semiring M.

Let x+y, x ∈ Mμ. Then μ(x+y) = μ(0), μ(x) = μ(0) since μ is a fuzzy k−ideal of Γ-semiring M. We have μ(y) ≥ min{μ(x + y), μ(x)} = min{μ(0), μ(0)} = μ(0), μ(y) ≥ μ(0) and μ(y) ≤ μ(0) ⇒ μ(y) = μ(0) ⇒ y ∈ Mμ.

Hence Mμ. is a k−ideal of Γ-semiring M. □

Proof of the following theorem follows from Theorems 3.45 and 3.46.

Theorem 3.47. Let M be a Γ-semiring and μ be a fuzzy subset of Γ-semiring M with |Im(μ)| = 2 and μ(0) = 1. Then Mμ is a k−ideal of Γ-semiring M if and only if μ is a fuzzy k−ideal of Γ-semiring M.

Theorem 3.48. If μ is a fuzzy 2−prime ideal and a fuzzy k−ideal of Γ-semiring M then μ is a fuzzy prime ideal of Γ-semiring M.

Proof. Suppose μ is a fuzzy 2−prime ideal and fuzzy k−ideal of Γ-semiring M, f and g be fuzzy ideals of Γ-semiring M such that fg ⊆ μ. As μ is a fuzzy k−ideal, by Lemma [3.19], since μ is a fuzzy 2−prime ideal of Γ-semiring M and are fuzzy k−ideals or Hence μ is a fuzzy prime ideal of Γ-semiring M. □

Theorem 3.49. Let μ be any fuzzy subset of Γ-semiring M, |Im(μ)| = 2 and μ(0) = 1. If Mμ is a 2−prime ideal of Γ-semiring M then μ is a fuzzy 2−prime ideal of Γ-semiring M.

Proof. Let μ be any fuzzy subset of Γ-semiring M, |Im(μ)| = 2, μ(0) = 1. Suppose Mμ is a 2−prime ideal of Γ-semiring M and Im(μ) = {t, 1}, t < 1. Clearly μ is a k−fuzzy ideal of Γ-semiring M. Let f and g be fuzzy k−ideals of Γ-semiring M such that fg ⊆ μ. Suppose that f ⊈ μ and g ⊈ μ. Then there exist x, y ∈ M f(x) > μ(x) and g(y) > μ(y). Clearly μ(x) = t = μ(y) ⇒ x, y ∉ Mμ. Since Mμ is a 2−prime ideal of Γ-semiring M, there exist x1 ∈ (x)k and y1 ∈ (y)k such that x1αy1 ∉ Mμ, α ∈ Γ. By Lemma 3.25, we have

Now

Therefore fg ⊈ μ, which is a contradiction. Hence μ is a fuzzy 2−prime ideal of Γ-semiring M. □

The proof of the following theorem follows from Theorems 3.48, 3.49 and Lemma 3.20.

Theorem 3.50. Let M be a Γ-semiring, μ be a fuzzy k−ideal of Γ-semiring M with |Im(μ)| = 2 and μ(0) = 1. Then Mμ is a 2−prime ideal of Γ-semiring M if and only if μ is a fuzzy prime ideal of Γ-semiring M.