South East Asian J. of Mathematics and Mathematical Sciences Vol. 18, No. 1 (2022), pp. 49-70 ISSN (Online): 2582-0850 ISSN (Print): 0972-7752 INTUITIONISTIC FUZZY CHARACTERISTIC IDEAL OF A Γ-RING P. K. Sharma and Hem Lata* Post- Graduate Department of Mathematics, D. A. V. College, Jalandhar - 144008, Punjab, INDIA E-mail : pksharma@davjalandhar.com *Lovely Professional University, Phagwara - 144001, Punjab, INDIA E-mail : goyalhema1986@gmail.com (Received: Aug. 23, 2021 Accepted: Feb. 21, 2022 Published: Apr. 30, 2022) Abstract: In this paper, we define the notion of intuitionistic fuzzy characteristic ideal (IFCI) of a Γ-ring which is analogue of a characteristic ideal in the ordinary ring theory and derive various new results. The correlation between the set of all automorphisms of Γ-ring and the corresponding automorphisms of its operator rings have been innovated. Then a one to one correlation between the set of all intuitionistic fuzzy characteristic ideals of Γ-ring and that of its operator ring has been constituted. This is used to obtain a similar bijection for characteristic ideals. Keywords and Phrases: Γ-ring, Intuitionistic fuzzy characteristic ideal, ΓAutomorphism. 2020 Mathematics Subject Classification: 03F55, 16D25, 08A72, 03G25. 1. Introduction The concept of a Γ-ring was first introduced by Nobusawa [9]. Barnes [5] weakened slightly the conditions in the definition of the Γ-ring in the sense of Nobusawa. Since then, many researchers have investigated various properties of this Γ-ring. Any ring can be regarded as a Γ-ring by suitably choosing Γ. Many fundamental results in ring theory have been extended to Γ-rings. R. Paul [13] 50 South East Asian J. of Mathematics and Mathematical Sciences studied various types of ideals in Γ-ring and the corresponding operator rings. The idea of intuitionistic fuzzy sets was first published by Atanassaov [3, 4], as a generalization of the notion of fuzzy set given by Zadeh [18]. Kim et al. in [8] considered the intuitionistic fuzzification of ideal of Γ-ring which were further studied by Palaniappan et al. in [10, 11]. Cho et al. in [6] and Palaniappan et al. in [12], studied intuitionistic fuzzy ideal and intuitionistic fuzzy prime ideal in Γ-nearrings. The notion of intuitionistic fuzzy bi-ideals in Γ-near-rings was introduced by Ezhilmaran et al. in [7]. Alhaleem et al. in [2] studied intuitionistic fuzzy normed subrings and ideals. The characteristic ideals and characteristic ideals of Γ-semigroups was studied by Sardar et al. in [14]. Aggarwal et al. in [1] studied some theorems on fuzzy prime ideals of Γ-rings. Sharma et al. in [15, 16, 17] studied extension of intuitionistic fuzzy ideals, intuitionistic fuzzy prime radical and intuitionistic fuzzy primary ideal and translational subset (ideals) in Γ-rings. The objective of this paper is to study the various properties of intuitionistic fuzzy characteristic ideal of a Γ-ring. We shall also investigate the relationship between the intuitionistic fuzzy characteristic ideal of a Γ-ring with its level cut sets. A connection between the set of all automorphisms of Γ-ring and the corresponding automorphisms of its operator rings will be characterized. Finally we will study the correspondence between the set of all intuitionistic fuzzy characteristic ideals of Γ-ring and the set of all intuitionistic fuzzy characteristic ideals of its operator ring. The structuring of the paper is as follows. In part 2 we recollect some groundwork for their use in the continuation of the development of the subject matter. In part 3 we set in motion of the notion of intuitionistic fuzzy characteristic ideal (IFCI) of Γ-ring M . With the help of an example we show that an intuitionistic fuzzy ideal of Γ-ring M need not be an intuitionistic fuzzy characteristic ideal. We also characterized intuitionistic fuzzy characteristic ideal with the help of its level cut Γ-ideals. In part 4 we inaugurate the notion of automorphism of operator rings of a Γ-ring and also the notion of corresponding automorphism in Γ-rings. While proving some more related results we establish a linkage among the set of all intuitionistic fuzzy characteristic ideals of Γ-ring and that of its operator ring. 2. Preliminaries Let us recall some definitions and results, which are necessary for the development of the paper. Definition 2.1. ([5, 9]) Let (M, +) and (Γ, +) be additive abelian groups. Then M is called a Γ-ring ( in the sense of Barnes [5]) if there exist mapping M ×Γ×M → M [image of (x, α, y) is denoted by xαy, x, y ∈ M, α ∈ Γ] satisfying the following con- Intuitionistic Fuzzy Characteristic Ideal of a Γ-ring 51 ditions: (1) xαy ∈ M . (2) (x + y)αz = xαz + yαz, x(α + β)y = xαy + xβy, xα(y + z) = xαy + xαz. (3) (xαy)βz = xα(yβz) for all x, y, z ∈ M and α, β ∈ Γ. The subset N of a Γ-ring M is a left ideal of M if N is an additive subgroup of M and M ΓN = {xαy|x ∈ M, α ∈ Γ, y ∈ N } is contained in N . Similarly, right ideal N ΓM of M can be defined. If N is both a left and a right ideal then N is ′ a two-sided ideal, or simply an ideal of M . A mapping f : M → M of Γ-rings is called a Γ-homomorphism [5] if f (x + y) = f (x) + f (y) and f (xαy) = f (x)αf (y) ′ for all x, y ∈ M, α ∈ Γ. When M = M , then a Γ-homomorphism is called a Γendomorphism, further a one to one Γ-endomorphism is called a Γ-automorphism. Throughout this study Aut(M ) will denote the set of all Γ-automorphisms of M . We now review some intuitionistic fuzzy logic concepts. We refer the reader to follow [3] and [4] for complete details. Definition 2.2. ([18]) A fuzzy set µ in X is a mapping µ : X → [0, 1]. Definition 2.3. ([3, 4]) An intuitionistic fuzzy set (IFS) A in X can be represented as an object of the form A = {< x, µA (x), νA (x) >: x ∈ X}, where the functions µA : X → [0, 1] and νA : X → [0, 1] denote the degree of membership (namely µA (x)) and the degree of non-membership (namely νA (x)) of each element x ∈ X to A respectively and 0 ≤ µA (x) + νA (x) ≤ 1 for each x ∈ X. It is shortly denoted by A(x) = (µA (x), νA (x)), for all x ∈ X. Proposition 2.4. ([3, 4]) If A and B are two intuitionistic fuzzy sets of X, then (i) A ⊆ B ⇔ µA (x) ≤ µB (x) and νA (x) ≥ νB (x), ∀x ∈ X; (ii) A = B ⇔ A ⊆ B and B ⊆ A, i.e., A(x) = B(x), for all x ∈ X. For any subset Y of X, the intuitionistic fuzzy characteristic function (IFCF) χY is an intuitionistic fuzzy set of X, defined as χY (x) = (1, 0), ∀x ∈ Y and χY (x) = (0, 1), ∀x ∈ X\Y . Let α, β ∈ [0, 1] with α + β ≤ 1. Then the set A(α,β) = {x ∈ X : µA (x) ≥ α and νA (x) ≤ β} is called the (α, β)-level cut subset of X with respect to IFS A. Further if f : X → Y is a mapping and A, B be respectively IFS of X and Y . Then the image f (A) is an IFS of Y and is defined as µf (A) (y) = Sup{µA (x) : f (x) = y}, νf (A) (y) = Inf {νA (x) : f (x) = y}, for all y ∈ Y and the inverse image f −1 (B) is an IFS of X and is defined as µf −1 (B) (x) = µB (f (x)), νf −1 (B) (x) = νB (f (x)), for all x ∈ X, i.e., f −1 (B)(x) = B(f (x)), for 52 South East Asian J. of Mathematics and Mathematical Sciences all x ∈ X. Also the IFS A of X is called f -invariant if f (x) = f (y) implies A(x) = A(y), where x, y ∈ X. Definition 2.5. ([8]) Let A be an IFS of a Γ-ring M . Then A is called an intuitionistic fuzzy ideal (IFI) of M if for all m, n ∈ M, α ∈ Γ, the following are satisfied (i) µA (m − n) ≥ µA (m) ∧ µA (n); (ii) µA (mαn) ≥ µA (m) ∨ µA (n); (iii) νA (m − n) ≤ νA (m) ∨ νA (n); (iv) νA (mαn) ≤ νA (m) ∧ νA (n). Example 2.6. Let D be a division ring matrices of the type  a 0 with unity 1 and M be a set of (2 × 2) b 0  where, a, b ∈ D. Take Γ = set of matrices of M with translation of interchanging of row 1 and row 2, then M is a Γ-ring. It is easy to see that the set J of all (2 × 2) matrices of the type   0 a 0 0 where, a ∈ D, is a Γ-ideal of M . Let A = (µA , νA ) be an IFS of M defined by ( ( 1, if x ∈ J 0, if x ∈ J µA (x) = ; νA (x) = 0.5, if x ∈ /J 0.3, if x ∈ / J. Then it is easy to verify that A is an IFI of Γ-ring M . Theorem 2.7. ([8]) Let K be a non-void subset of a Γ-ring M . Then K is Γ-ideal of M iff intuitionistic fuzzy characteristic function χK is an intuitionistic fuzzy ideal of M . 3. Intuitionistic Fuzzy Characteristic Ideal of Γ-Ring Definition 3.1. Let A be an IFS in a Γ-ring M and f : M → M be a Γendomorphism, then Af is an IFS on M defined as Af (x) = A(f (x)), for all x ∈ M , i.e., µAf (x) = µA (f (x)) and νAf (x) = νA (f (x)), for all x ∈ M . Theorem 3.2. Let A be an IFI of Γ-ring M and f be a Γ-endomorphism, then Intuitionistic Fuzzy Characteristic Ideal of a Γ-ring 53 Af is also an IFI of M . Proof. Let A be an IFI of Γ-ring M . Let x, y ∈ M, α ∈ Γ. Then µAf (x − y) = = ≥ = µA (f (x − y)) µA (f (x) − f (y)) µA (f (x)) ∧ µA (f (y)) µAf (x) ∧ µAf (y). Thus µAf (x − y) ≥ µAf (x) ∧ µAf (y). Similarly, we can prove νAf (x − y) ≤ νAf (x) ∨ νAf (y). Also, µAf (xαy) = = ≥ = µA (f (xαy)) µA (f (x)αf (y)) µA (f (x)) ∨ µA (f (y)) µAf (x) ∨ µAf (y). i.e., µAf (xαy) ≥ µAf (x) ∨µAf (y). Similarly, we can prove νAf (xαy) ≤ νAf (x) ∧µAf (y). Hence Af is an IFI of Γ-ring M . Definition 3.3. A Γ-ideal K of M is said to be characteristic ideal if f (K) = K for all f ∈ Aut(M ). Definition 3.4. An IFI A of Γ-ring M is said to be an IFCI if Af (x) = A(x), ∀x ∈ M and for all f ∈ Aut(M ), i.e., µAf (x) = µA (x) and νAf (x) = νA (x) for all x ∈ M and for all f ∈ Aut(M ). Example 3.5. Consider the Γ-ring M , where M = Z, the ring of integers and Γ = 2Z, the ring of even integers and xγy denote the usual product of integers x, y ∈ M, γ ∈ Γ. Let A = (µA , νA ) be an intuitionistic fuzzy subset of M defined by ( ( 1, if x is even integer 0, if x is even integer µA (x) = ; νA (x) = 0.5, if x is odd integer 0.3, if x is odd integer. Then it is easy to verify that A is an IFCI of Γ-ring M . Example 3.6. Consider the Γ-ring M , where M = {[aij ] : aij ∈ Z, i = 1, 2, j = 1, 2, 3}, the set of (2 × 3) matrices and Γ = {[aij ] : aij ∈ Z, i = 1, 2, 3, j = 1, 2}, the set of (3 × 2) matrices whose entries are from the ring of integers Z. Let 54 South East Asian J. of Mathematics and Mathematical Sciences A = (µA , νA ) be an IFS of M defined by ( (0.7, 0.2), if aij = 0, ∀i, j A([aij ]) = (0.3, 0.5), if aij 6= 0 for atleast one i and j Then it is easy to verify that A is an IFCI of Γ-ring M . Example 3.7. Consider M = Z2 × Z2 = {(0, 0), (1, 0), (0, 1), (1, 1)}, Γ = {(0, 0), (1, 1)} and K = Z2 × {0} = {(1, 0), (0, 0)}, where Z2 be the ring of integers modulo 2. Clearly, M and Γ are additive abelian groups and that M is Γ-ring. Also, here K is Γ-ideal of M . Consider the IFS A defined on M as ( ( 1, if x ∈ K 0, if x ∈ K µA (x) = ; νA (x) = 0.5, if x ∈ /K 0.3, if x ∈ / K. Then it is easy to check that A is an IFI of Γ-ring M , but it is not an IFCI, as there exists a Γ-automorphism f : M → M defined by f (x, y) = (y, x), for all (x, y) ∈ M such that Af ((x, y)) 6= A((x, y)), for all (x, y) ∈ M . For example: Af ((1, 0)) = (0.5, 0.3) 6= (1, 0) = A((1, 0)). Theorem 3.8. Let A be an IFCI of Γ-ring M . Then for each α, β ∈ [0, 1] such that α + β ≤ 1 the level cut set A(α,β) is a characteristic ideal of Γ-ring M . Proof. Assume that A be an IFCI of Γ-ring M . It is sufficient to show that f (A(α,β) ) = A(α,β) for all α, β ∈ [0, 1] such that α + β ≤ 1. Let x ∈ A(α,β) . Since A be an IFCI of Γ-ring M , we have µAf (x) = µA (x) ≥ α and νAf (x) = νA (x) ≤ β implies µA (f (x)) ≥ α and νA (f (x)) ≤ β, i.e., f (x) ∈ A(α,β) . Thus f (A(α,β) ) ⊆ A(α,β) . For the reverse inclusion, let y ∈ A(α,β) and let x ∈ M be such that f (x) = y. Then µA (x) = µAf (x) = µA (f (x)) = µA (y) ≥ α. Similarly, we can prove νA (x) ≤ β implies x ∈ A(α,β) and so y = f (x) ∈ f (A(α,β) ) gives that A(α,β) ⊆ f (A(α,β) ). Thus f (A(α,β) ) = A(α,β) . Hence A(α,β) is a characteristic ideal of Γ-ring M . The following lemma is obvious, and we omit the proof. Lemma 3.9. Let A be an IFI of Γ-ring M and let x ∈ M . Then A(x) = (α, β) if and only if x ∈ A(α,β) and x ∈ / A(p,q) for all p > α and q < β. Now we give the converse of the Theorem (3.8). Theorem 3.10. Let A be an IFI of Γ-ring M . If for each α, β ∈ [0, 1] such that α + β ≤ 1 the level cut set A(α,β) is a characteristic ideal of M , then A is an IFCI of Γ-ring M . Intuitionistic Fuzzy Characteristic Ideal of a Γ-ring 55 Proof. Let A be an IFI of Γ-ring M . Let x ∈ M , f ∈ Aut(M ) and A(x) = (α, β). By Lemma (3.9), x ∈ A(α,β) and x ∈ / A(p,q) for all p > α and q < β. From hypothesis it follows that f (A(α,β) ) = A(α,β) . Thus f (x) ∈ f (A(α,β) ) = A(α,β) , and so µA (f (x)) ≥ α, νA (f (x)) ≤ β. Let µA (f (x)) = p and νA (f (x)) = q and we assume that p > α and q < β. Then f (x) ∈ A(p,q) = f (A(p,q) ). Since f is one to one implies x ∈ A(p,q) . This is a contradiction. Hence µAf (x) = µA (f (x)) = α = µA (x) and νAf (x) = νA (f (x)) = β = νA (x), showing that A is an IFCI of Γ-ring M . Theorem 3.11. A non-empty subset K of a Γ-ring M is a characteristic ideal of M iff its IFCF χK is an IFCI of Γ-ring M . Proof. Let K be a characteristic ideal of Γ-ring M . Then by definition f (K) = K, ∀f ∈ Aut(M ). Let χK be the IFCF with respect to K. Then by Theorem (2.7) χK be an IFI of Γ-ring M . Also, If x ∈ K then ∀f ∈ Aut(M ), we have f (x) ∈ f (K) = K and so χK (f (x)) = (1, 0) = χK (x). If x ∈ / K then ∀f ∈ Aut(M ), we have f (x) ∈ / f (K) = K and so χK (f (x)) = (0, 1) = χK (x). Thus we see that χK (f (x)) = χK (x), ∀x ∈ M, ∀f ∈ Aut(M ), i.e., µχf (x) = K µχK (x) and νχf (x) = νχK (x), ∀x ∈ M, ∀f ∈ Aut(M ). Hence χK is an IFCI of K Γ-ring M . Conversely, let us suppose that χK be an IFCI of Γ-ring M . Then by Theorem (2.7) K is an Γ-ideal of M . So we need only to show that f (K) = K.∀f ∈ Aut(M ). Let f ∈ Aut(M ) and x ∈ K, then µχf (x) = µχK (x) = 1 and νχf (x) = νχK (x) = 0 K K implies µχK (f (x)) = 1 and νχK (f (x)) = 0 implies f (x) ∈ K. Thus we obtain f (K) ⊆ K, for all f ∈ Aut(M ). Since f ∈ Aut(M ) implies f −1 ∈ Aut(M ) and so f −1 (K) ⊆ K. Hence K ⊆ f (K) and so f (K) = K, i.e., K is characteristic ideal of M. 4. Operator rings and corresponding intuitionistic fuzzy ideals of Γ-ring Definition 4.1. ([6, 9]) Let M be a Γ-ring. Let us signify a relation σ on M × Γ as follows: (x, α)σ(y, β) if and only if xαm = yβm, ∀m ∈ M and γxα = γyβ, ∀γ ∈ Γ. Thus σ is an equivalence relation on M × Γ. Set [x, α] be the equivalence class containing (x, α). Let L = {[x, α] : x ∈ M, α ∈ Γ}. Then L is a ring with respect to the compositions 56 South East Asian J. of Mathematics and Mathematical Sciences [x, α] + [y, + y, α] ; [x, α] P+ [x, β] = [x, α + β] ; P α] = [xP i,j [xi αi yj , βj ]. j [yj , βj ] = i [xi , αi ] This ring L is called the left operator ring of Γ-ring M . Dually the right operator ring R of Γ-ring M is formed where the compositions on R are defined as: Remark 4.2. [α, x] +P [β, x] = [αP+ β, x]; [α, P x] + [α, y] = [α, x + y]; i [αi , xi ] j [βj , yj ] = i,j [αi , xi βj yj ]. P P (1) If there 1L = i [ei , δi ] ∈ L ( or 1R = i [γP i , ai ] ∈ R) such P exists an element P xγ a = x) for all x ∈ M then e δ x = x (resp. that i ] (resp. i i i [ei , δ i i i i P P called the left (resp. right) unity of M . Also 1L = i [ei , δi ] i [γi , ai ]) isP (resp. 1R = i [γi , ai ]) is the unity of L ( resp. R). P P (2) If we define a mapping L×M → M by ( i [xi , αi ], y) → i xi αi y, then we can show that the above mapping is well defined and M is a left L-module, and we call L the left operator ring of the Γ-ring M . Similarly, we can construct a right operator ring R of M so that M is a right R-module. Let M be a Γ-ring with left operator ring L. For P ⊆ L and Q ⊆ M , we define ′ P = {x ∈ M : [x, α] ∈ P, ∀α ∈ Γ} and Q+ = {[x, α] ∈ L : xαy ∈ Q, ∀y ∈ M }. Similarly, if M is a Γ-ring with and right operator ring R. For P ⊆ R and Q ⊆ M , we define ′ P ∗ = {x ∈ M : [α, x] ∈ P, ∀α ∈ Γ} and Q∗ = {[α, x] ∈ R : yαx ∈ Q, ∀y ∈ M } Then in [6], it was shown that if P (resp. Q) is a right ideal of L (resp. M ), ′ then P + (resp. Q+ ) is a right ideal of M (resp. L) and there exists an inclusion ′ preserving mapping Q → Q+ . Also if P (resp. Q) is a left ideal of R (resp. M ), ′ then P ∗ (resp. Q∗ ) is a left ideal of M (resp. R) and there exists an inclusion ′ preserving mapping Q → Q∗ . + Definition 4.3. Let M be a Γ-ring and L be the left operator ring of M . Then the bijection f : L → L is said to be automorphism if 1. f ([x, α] + [y, α]) = f ([x, α]) + f ([y, α]) and f ([x, α] + [x, β]) = f ([x, α]) + f ([x, β]), P P P P 2. f ( i [xi , αi ] j [yj , βj ]) = f ( i [xi , αi ])f ( j [yj , βj ]), P P P 3. f ( i [ei , δi ]) = i [ei , δi ], if i [ei , δi ] is the left unity of M , Intuitionistic Fuzzy Characteristic Ideal of a Γ-ring 4. f ( P i [ai , γi ]) = P i [ai , γi ], if P i [ai , γi ] 57 is the right unity of M . Similarly we can define the automorphism on the right operator ring R of the Γ-ring M . Proposition 4.4. ([10]) Every left (or right) ideal of Γ-ring M defines a left (or right) ideal of the right operator ring R and conversely. Definition 4.5. Let L and R be respectively be the left and right operator ring of Γ-ring M . Then for any fixed IFS A of L (or R) and for any fixed IFS B of M ′ ′ we define intuitionistic fuzzy sets A+ , A∗ of M and B + of L, B ∗ of R by µA+ (x) = Infα∈Γ (µA ([x, α])) and νA+ (x) = Supα∈Γ (µA ([x, α])), where x ∈ M . µA∗ (x) = Infα∈Γ (µA ([α, x])) and νA∗ (x) = Supα∈Γ (µA ([α, x])), where x ∈ M . P P P ′( x α m)) and ν (µ ( µB +′ ( i [xi , αi ]) = Inf i i m∈M B + i [xi , αi ]) i B P = Supm∈M (µB ( i xi αi m)), where [xi , αi ] ∈ L. P P P ′ µB ∗′ ( i [αi , xi ]) = Inf Pm∈M (µB ( i mαi xi )) and νB ∗ ( i [αi , xi ]) = Supm∈M (µB ( i mαi xi )), where [αi , xi ] ∈ R. Proposition 4.6. Let M be a Γ-ring and L be the left operator ring of M and A is an IFI of L. Then A+ is an IFI of M . Proof. Let A is an IFI of L. Then µA (0L ) = 1, νA (0L ) = 0. Now µA+ (0M ) = Infα∈Γ (µA ([0M , α])) = Infα∈Γ (µA (0L )) = 1. Similarly, we can show that νA+ (0M ) = 0. So A+ is non-empty. Let x, y, m ∈ M, α, β ∈ Γ be any elements, then we have µA+ (x − y) = = ≥ = = Infα∈Γ (µA ([x − y, α])) Infα∈Γ (µA ([x, α] − [y, α])) Infα∈Γ {µA ([x, α]) ∧ µA ([y, α])} Infα∈Γ (µA ([x, α])) ∧ Infα∈Γ (µA ([y, α])) µA+ (x) ∧ µA+ (y). Thus µA+ (x − y) ≥ µA+ (x) ∧ µA+ (y). Similarly, we can prove νA+ (x − y) ≤ νA+ (x) ∨ νA+ (y). Also, µA+ (xβy) = = ≥ = = Infα∈Γ (µA ([xβy, α])) Infα∈Γ (µA ([x, β][y, α])) Infα∈Γ (µA ([x, β]))[ and ≥ Infα∈Γ (µA ([y, α]))] Infβ∈Γ (µA ([x, β])) ∨ Infα∈Γ (µA ([y, α])) µA+ (x) ∨ µA+ (y). 58 South East Asian J. of Mathematics and Mathematical Sciences Thus µA+ (xαy) ≥ µA+ (x) ∨ µA+ (y). Similarly, we can prove νA+ (xαy) ≤ νA+ (x) ∧ νA+ (y). Hence A+ is an IFI of M . Proposition 4.7. Let M be a Γ-ring and L be the left operator ring of M and B ′ is an IFI of M . Then B + is an IFI of L. Proof. Let B is an IFI of M . Then µB (0M ) = 1, νB (0M ) = 0. Now µB +′ ([0M , α]) = Infm∈M (µB (0M αm)) = µB (0M ) = 1. Similarly, we can show ′ thatP νB +′ ([0M , P α]) = 0. So B + is non-empty. Let i [xi , αi ], j [yj , βj ] ∈ L , m ∈ M, αi , βj ∈ Γ be any elements, then we have µ B +′ ( X [xi , αi ] − i X [yj , βj ]) = Infm∈M (µB ( j X xi α i m − i ≥ Infm∈M {µB ( X yj βj m)) j X xi αi m) ∧ µB ( i = (Infm∈M (µB ( X X yj βj m)} j xi αi m))) ∧ (Infm∈M (µB ( i = µ B +′ ( X X yj βj m))) j [xi , αi ]) ∧ µB +′ ( X [yj , βj ]). j i P Thus µB +′ ( i [xi , αi ] − j [yj , βj ]) ≥ µB +′ ( i [xi , αi ]) ∧ µB +′ ( j [yj , βj ]). Similarly, P P P P we can show νB +′ ( i [xi , αi ] − j [yj , βj ]) ≤ νB +′ ( i [xi , αi ]) ∨ µB +′ ( j [yj , βj ]) Also. µ B +′ ( X i [xi , αi ] P P P X j [yj , βj ]) = µB +′ ( X [xi αi yj , βj ]) i,j = Infm∈M (µB ( X xi αi yj βj m)) i,j = Infm∈M (µB ( X (xi αi )(yj βj m))) i,j = Infm′ ∈M (µB ( j X ′ ′ xi αi mj ))[ where mj = yj βj m ∈ M ] i,j = Infm′ ∈M [µB ( j X ′ x i α i m1 + i ≥ Infm′ ∈M [∨j µB ( j i X ′ xi αi mj )] i = ∨j [Infm′ ∈M ( j = ∨j [µB +′ ( X i ′ xi αi mj )] i X X [xi , αi ])] ′ xi αi m2 + ...)] Intuitionistic Fuzzy Characteristic Ideal of a Γ-ring = µB +′ ( X 59 [xi , αi ]). i P P P Also, we can prove that µB +′ ( i [xi , αi ] j [yj , βj ]) ≥ µB +′ ( j [yj , βj ]). Thus we have P P P P µB +′ ( i [xi , αi ] j [yj , βj ]) ≥ µB +′ ( i [xi , αi ]) ∨ µB +′ ( j [yj , βj ]). Similarly, we can prove ′ P P P P νB +′ ( i [xi , αi ] j [yj , βj ]) ≤ νB +′ ( i [xi , αi ]) ∧ νB +′ ( j [yj , βj ]). Hence B + is an IFI of L. Similarly, we can prove the following propositions. Proposition 4.8. Let M be a Γ-ring and R be the right operator ring of M and A an IFI of R. Then A∗ an IFI of M . Proposition 4.9. Let M be a Γ-ring and R be the right operator ring of M and ′ B an IFI of M . Then B ∗ an IFI of R. Theorem 4.10. Let M be a Γ-ring with unities and L be its left operator ring. ′ Then ∃ an inclusion preserving one to one map A → A+ between the set of all intuitionistic fuzzy ideals of M and the set of intuitionistic fuzzy ideals of L. ′ Proof. First we show that ((A+ ) )+ = A, where A is an IFI of M . Let x ∈ M . Then µ((A+ )′ )+ (x) = = ≥ = Infα∈Γ (µ(A+ )′ ([x, α])) Infα∈Γ [Infm∈M (µA (xαm))] Infα∈Γ [Infm∈M (µA (x))] µA (x). Thus µ((A+ )′ )+ (x) ≥ µA (x). Similarly, we can prove ν((A+ )′ )+ (x) ≤ νA (x). Thus ′ A ⊆P ((A+ ) )+ . P Let i [γi , ai ] be the right unity of M . Then i xγi ai = x, for all x ∈ M . Now, µA (x) = µA ( X xγi ai ) i ≥ ≥ = = Infi [µA (xγi ai )] Infγ∈Γ [Infm∈M (µA (xγm))] Infγ∈Γ (µ(A+ )′ ([x, γ])) µ((A+ )′ )+ (x). 60 South East Asian J. of Mathematics and Mathematical Sciences ′ Similarly, we can prove νA (x) ≤ ν((A+ )′ )+ (x). So ((A+ ) )+ ⊆ A. Hence A = ′ ((A+ ) )+ . Again, let A be an IFI of L. Now, X X µ((A+ )+ )′ ( [xi , αi ]) = Infm∈M (µA+ ( xi αi m)) i i = Infm∈M [Infβ∈Γ (µA ([ X xi αi m, β]))] i = Infm∈M [Infβ∈Γ (µA ( X [xi , αi ][m, β]))] i ≥ µA ( X [xi , αi ]). i P P Thus µ((A+ )+ )′ ( i [xi , αi ] ≥ µA ( i [xi , αi ]). Similarly, we can prove P P + + ′ ′ ν((A+P )+ ) ( i [xi , αi ]). So A ⊆ ((A ) ) . i [xi , αi ] ≤ νA ( Let j [aj , γj ] be the right unity of M , then X X X µA ( [xi , αi ]) = µA ( [xi , αi ] [aj , γj ]) i i ≥ ∧j [µA ( j X [xi , αi ][aj , γj ])]] i ≥ Infm∈M [Infγ∈Γ (µA ([xi , αi ][aj , γj ]))] X = µ((A+ )+ )′ ( [xi , αi ]). i P P Thus µA ( i [xi , αi ]) ≥ µ((A+ )+ )′ ( i [xi , αi ]). Similarly, we can prove P P ′ ′ νA ( i [xi , αi ]) ≤ ν((A+ )+ )′ ( i [xi , αi ]) and so ((A+ )+ ) ⊆ A and hence A = ((A+ )+ ) . ′ Thus the correspondence A → A+ is a bijection. Now P let A1 , A2 be intuitionistic fuzzy ideals of M such that A1 ⊆ A2 . Then for all i [xi , αi ] ∈ L, we have X X µ +′ ( [xi , αi ]) = Infm∈M (µA1 ( xi αi m)) A1 i i ≤ Infm∈M (µA2 ( X xi αi m)) i = µ A+ 2 ′ ( X [xi , αi ]). i P P P Thus µ +′ ( i [xi , αi ]) ≤ µ +′ ( i [xi , αi ]). Similarly, we can show ν +′ ( i [xi , αi ]) ≥ A1 A2 A ′ ′ P1 + + ν +′ ( i [xi , αi ]). Thus A1 ⊆ A2 . Similarly we can show that if A1 , A2 are intuA2 Intuitionistic Fuzzy Characteristic Ideal of a Γ-ring 61 ′ + + itionistic fuzzy ideals of L such that A1 ⊆ A2 , then A+ is 1 ⊆ A2 . Hence A → A an inclusion preserving one to one map. Similarly, we can prove the following theorem. Theorem 4.11. Let M be a Γ-ring with unities and R be its right operator ring. ′ Then ∃ an inclusion preserving one to one map B → B ∗ between the set of all IFIs of M and the set of IFIs of R. Proposition 4.12. Let K be an ideal of the left operator ring L of a Γ-ring M . Then (χK )+ = χK + , where χK denote the IFCF of K. Proof. Let x ∈ K + . Then [x, α] ∈ K for all α ∈ Γ. This mean Infα∈Γ (µχK ([x, α])) = 1 and Supα∈Γ (νχK ([x, α])) = 0. Also µχK + (x) = 1 and νχK + (x) = 0. Thus Infα∈Γ (µχK ([x, α])) = µχK + (x) and Supα∈Γ (νχK ([x, α])) = νχK + (x), ∀x ∈ K + , i.e., (χK )+ (x) = χK + (x), ∀x ∈ K + . Now suppose x ∈ / K + . Then ∃β ∈ Γ such that [x, β] ∈ / K. Hence µχK ([x, β]) = 0, νχK ([x, β]) = 1 and so Infα∈Γ (µχK ([x, α])) = 0 and Supα∈Γ (νχK ([x, α])) = 1. Thus Infα∈Γ (µχK ([x, α])) = µχK + (x) and Supα∈Γ (νχK ([x, α])) = νχK + (x), ∀x ∈ / / K + . Hence (χK )+ = χK + . K + , i.e., (χK )+ (x) = χK + (x), ∀x ∈ By applying similar argument as above we deduce the following Lemma. Lemma 4.13. Let K be an ideal of a Γ-ring M and L be the left operator ring of ′ M . Then (χK )+ = χK +′ . ′ P P + Proof. Let i [xi , αi ] ∈ K P . Then i xi αi m ∈ K, ∀m ∈ M P. This means P Infm∈M µχK ( i xi αi m) = 1P and Supm∈M νχK ( i xi αi m) = 0, i.e., µ(χ )+′ ( i [xi , αi ]) = 1 and ν(χ )+′ ( i [xi , αi ]) = 0. K K P P Also µ(χ +′ ) ( i [xi , αi ]) = 1 and ν(χ +′ ) ( i [xi , αi ]) = 0. Thus we have K P P P P K µ(χ +′ ) ( i [xi , αi ]) = µ(χ )+′ ( i [xi , αi ]) and ν(χ +′ ) ( i [xi , αi ]) = ν(χ )+′ ( i [xi , αi ]). K K K K ′ P P + So (χK ) ( i [xi , αi ]) = (χK +′ )( i [xi , αi ]). ′ P P / K, ∀m ∈ M . P Then i xi αi m ∈ / K+ . P Let i [xi , αi ] ∈ and Supm∈M νχK ( i xi αi m) = 1, This means P Infm∈M µχK ( i xi αi m) = 0P i.e., µ(χ )+′ ( i [xi , αi ]) = 0 and ν(χ )+′ ( i [xi , αi ]) = 1. K K P P Also µ(χ +′ ) ( i [xi , αi ]) = 0 and ν(χ +′ ) ( i [xi , αi ]) = 1. Thus we have K P P P P K µ(χ +′ ) ( i [xi , αi ]) = µ(χ )+′ ( i [xi , αi ]) and ν(χ +′ ) ( i [xi , αi ]) = ν(χ )+′ ( i [xi , αi ]). K K K K ′ P P + ′ So (χK ) ( i [xi , αi ]) = (χK + )( i [xi , αi ]). ′ Thus from both the cases we get (χK )+ = χK +′ . Remark 4.14. By drawing an analogy we can deduce results similar to the above 62 South East Asian J. of Mathematics and Mathematical Sciences Lemmas for right operator ring R of the Γ-ring M . Theorem 4.15. Let M be a Γ-ring with unities. Then ∃ an inclusion preserving one to one between the set of all ideals of M and that of its left operator ring L via ′ the mapping K → K + . ′ Proof. Let φ : K → K + be the mapping. This is actually a mapping follows ′ ′ from Proposition (4.9). Now let φ(K1 ) = φ(K2 ). Then K1+ = K2+ . This implies ′ χ +′ = χ +′ (where χK is the IFCF of K). Hence by Lemma (4.13), (χK1 )+ = K1 K2 + ′ (χK2 ) . This together with Theorem (4.10) gives χK1 = χK2 , hence K1 = K2 . Consequently φ is one to one. Let K be an ideal of L. Then its IFCF χK is an IFI of L. Hence by Theo′ rem (4.10), ((χK )+ )+ = χK . This implies that χ(K + )+′ = χK [ by Lemma (4.12) ′ and (4.13)]. Hence (K + )+ = K, i.e., φ(K + ) = K. Now since K + is an ideal of M , it follows that φ is onto. Let K1 , K2 be two ideals of M with K1 ⊆ K2 . ′ ′ Then χK1 ⊆ χK2 . Hence by Theorem (4.10) we see that (χK1 )+ ⊆ (χK2 )+ , i.e., ′ ′ χ +′ ⊆ χ +′ [ by Lemma (4.13)] which gives K1+ ⊆ K2+ . K1 K2 Remark 4.16. Now by using a similar argument as above with the help of Lemma dual to Lemmas (4.12) and (4.13), Remark (4.14) and Theorem (4.12) we can de′ duce that ()∗ is an inclusion preserving one-to-one map (with ()∗ as above) between the set of all ideals of M and that of its right operator ring R. Definition 4.17. Let M be a Γ-ring and L be its left operator ring. Then for ′ ′ P P f ∈ Aut(M ), we define f + : L → L by f + ( i [xi , αi ]) = i [f (xi ), αi ]. ′ We first the map f + is well-defined. P show thatP = yj βj m, ∀m ∈ Suppose Pi [xi , αi ] = Pj [yj , βj ], then [xi , αi ] = [yP j , βj ], so, xi αi mP M . Thus i xi αi m = j yj βj m. This implies f ( i xi αi m) = f ( j yj βm), ∀m ∈ M. ′ Now for a ∈ M , we have f (xi )αi a = f (xi )αi f (a ) [As f is onto so there exists ′ ′ ′ ′ ′ a ∈ M such that f (a ) = a] = f (xi αi a ) = f (yj βj a ) = f (yj )βj f (a P) = f (yj )βj a. This implies f (xi )αi a = f (yj )βj a. So [f (xi ), αi ] = [f (yj ), βj ] ⇒ i [f (xi ), αi ] = ′ ′ P ′ P P + + + is j [yj , βj ]). Therefore the map f i [xi , αi ]) = f ( j [f (yj ), βj ] . Hence f ( well-defined. Proposition 4.18. Let M be a Γ-ring and L be its left operator ring. Let f ∈ ′ Aut(M ). Then f + ∈ Aut(L). Proof. Let f ∈ Aut(M ) and [x, α], [y, α], [x, β] ∈ L. Then Intuitionistic Fuzzy Characteristic Ideal of a Γ-ring ′ 63 ′ f + ([x, α] + [y, α]) = f + ([x + y, α]) = [f (x + y), α] = [f (x) + f (y), α] = [f (x), α] + [f (y), β] ′ ′ f + ([x, α] + [x, β]) = f + ([x, α + β]) = [f (x), α + β] = [f (x), α] + [f (x), β]. ′ f+ ( X i [xi , αi ] X ′ [yj , βj ]) = f + ( X [xi αi yj , βj ]) i,j j = X [f (xi αi yj ), βj ] i,j = X [f (xi )αi f (yj ), βj ] i,j = X [f (xi ), αi ] i ′ = f+ ( ′ [f (yj ), βj ] j X i ′ X ′ [xi , αi ])f + ( X [yj , βj ] j ′ Hence f + is an endomorphism of L. As f + is well-defined implies f + is one to one map. P P ′ ′ ′ Further, let i [xi , αi ] ∈ L. Then ∃, xi ∈ M such that f (xi ) = xi . So i [xi , αi ] ∈ L ′ ′ P P P ′ ′ such that f + ( i [xi , αi ]) = i [f (xi ), αi ] = i [xi , αi ]. Consequently, f + is onto. ′ P P + [e , δ ]. Then for any α ∈ Γ, we have f Suppose L has the left unity i i i i [ei , αi ] i P P P = i [f (ei ), αi ] = i [ei , αi ]. Again if M has the right unity i [γi , ai ]. Then for any ′ ′ P P P + [γ , α ]. Hence f [f (γ ), α ] = ∈ Aut(L). [γ , α ] = αi ∈ Γ, we have f + i i i i i i i i i We use the Remark (4.2)(ii) to frame the following precision and also to demonstrate the subsequent Propositions. P Definition 4.19. Let M be a Γ-ring with right unity i [γi , ai ] and L Pbe its left op+ + erator ring. Then for f ∈ Aut(L), we set f : M → M by f (x) = i f ([x, γi ])ai . We first show that the map f + is well-defined P Let x, y ∈ M, γi , βi ∈ Γ be such that f + (x) = f + (y), then i f ([x, γi ])ai = P i f ([y, P Pγi ])ai ([y, γi ])ai , γP ⇒ Pi [f ([x, γi ])aP i] i , γi ] = i [f P f ([y, γi ]) i [aP ⇒ Pi f ([x, γi ]) P i , γi ] iP i [ai , γi ] = f ([y, γ ])f ( [a , γ ]) = f ([x, γ ])f ( ⇒ P i P i [ai , γi ]) [ Using Definition (4.3)] i i i i Pi Pi = f ( i [y, γi ] i [ai , γi ]) ⇒ f (Pi [x, γi ] i [ai , γi ])P = f ( i [yγi ai , γi ]) ⇒ fP( i [xγi ai , γi ])P γi ] = i [yγP ⇒ Pi [xγi ai ,P i ai , γi ] [ Since P f is one to one ] ⇒ i [x, γi ] i [ai , γi ] = i [y, γi ] i [ai , γi ] 64 South East Asian J. of Mathematics and Mathematical Sciences P P [y, γi ] ⇒ xγP ⇒ i [x, γi ] = i [y, γi ] ⇒ [x, γi ] = P i m = yγi m, ∀m ∈ M . In particular, take m = ai , we get i xγi ai = i yγi ai ⇒ x = y. Hence f + is well-defined. P Proposition 4.20. Let M be a Γ-ring with right unity i [γi , ai ] and L be its left operator ring. Assume f ∈ Aut(L), then f + ∈ Aut(M ). Proof. Let x, y ∈ M, α ∈ Γ. Then f + (x + y) = X f ([x + y, γi ])ai i = X f ([x, γi ] + [y, γi ])ai i = X (f ([x, γi ])ai + f ([y, γi ])ai )) i = X f ([x, γi ])ai + i X f ([y, γi ])ai i = f + (x) + f + (y). X X f + (xαy) = f ([xαy, γi ])ai = f ([x, α][y, γi ])ai i = i X f ([x, α]) i = X X f ([y, γi ])ai = X f ([x, γi ][ai , α]) X f ([x, γi ]) X X [ai , α] i f ([x, γi ])ai )α( f ([y, γi ])ai = i X f ([x, γi ]) i X f ([y, γi ])ai = i X X f ([y, γi ])ai i i i = ( f ([xγi ai , α]) i i i = X X i X f ([x, γi ])ai α i f ([y, γi ])ai ) f ([ai , α]) X f ([y, γi ])ai i X f ([y, γi ])ai i i = f + (x)αf + (y). Hence f + is an endomorphism of M . As f + is well-defined implies that f + is one to one map. P Further, let y ∈ M . Since f : L → L is onto, ∃ i [x, γi ] ∈ L such that P P f ( i [x, γi ]) = i [y, γi ]. f + (x) = X i = X i f ([x, γi ])ai = X f ([xγi ai , γi ])ai i f ([x, γi ][ai , γi ])ai = X i f ([x, γi ]) X i f ([ai , γi ])ai Intuitionistic Fuzzy Characteristic Ideal of a Γ-ring = X [y, γi ] i = X X [ai , γi ]ai = i i = [y, γi ][ai , γi ]ai i [yγi ai , γi ]ai = X X 65 X [y, γi ]ai i yγi ai = y. i P ] is the left unity of M then HencePf + is onto. Again if i [ei , δiP P f + (e) = i f ([e, δi ])ai = i [e, δi ]ai = i eδi ai = e. Consequently, f + ∈ Aut(M ). P Proposition 4.21. Let M be a Γ-ring with left unity i [ei , δi ] and right unity ′ P + + [γ , a ] and L be its left operator ring. Assume f ∈ Aut(L), then (f ) = f. i i i ′ ′ + Proof. By Proposition (4.18), f ∈ Aut(L) hence by Proposition (4.20), (f + )+ ∈ ′ ′ P P + + + [x, γ ])a = Aut(M ). Let x ∈ M . Then (f ) (x) = f ( i i i [f (x), γi ]ai i P = i f (x)γi ai = f (x). ′ Hence (f + )+ = f . P Proposition 4.22. Let M be a Γ-ring with left unity i [ei , δi ] and right unity ′ P + + [γ , a ] and L be its left operator ring. Let f ∈ Aut(M ). Then (f ) = f. i i i Proof. By Proposition (4.20), f + ∈ Aut(M ) whence by Proposition (4.18), ′ P (f + )+ ∈ Aut(L). Let i [xi , αi ] ∈ L. Then X X ′ X (f + )+ ( [xi , αi ]) = [f + (xi ), αi ] = [f ([xi , γi ])ai , αi ] i i = i X f ([xi , γi ]) i = X X i f ([xi , γi ][ai , αi ]) = i = f( [ai , αi ] = i X i X [xi , αi ]). X f ([xi , γi ])f ( X f ([xi γi ai , αi ]) = [ai , αi ]) i X f ([xi , αi ]) i i ′ Hence (f + )+ = f . Theorem 4.23. Let M be a Γ-ring and L be its left operator ring. Then there exists a bijection between the set of all automorphisms of M and the set of all automorphisms of L. ′ Proof. Let us define the map φ : Aut(M ) → Aut(L) by φ(f ) = f + , ∀f ∈ Aut(M ). ′ ′ Consider f, g ∈ Aut(M ) such that φ(f ) = φ(g). Then f + = g + ′ P ′ P P P P ⇒ f + ( i [xi , αi ]) = g + ( i [xi , αi ]), ∀ i [xi , αi ] ∈ L ⇒ i [f (xi ), αi ] = i [g(xi ), αi ] 66 South East Asian J. of Mathematics and Mathematical Sciences ⇒ f (xi )αi m = g(xi )αi m, ∀m ∈ M, αi ∈ Γ. In particular, f (xi )γi ai = g(xi )γi ai ⇒ f (xi ) = g(xi ). So f = g. Hence φ is one to one. ′ Suppose f ∈ Aut(M ). Then by Proposition (4.20), f + ∈ Aut(M ). Now ′ φ(f + ) = f + = f (by Proposition (4.22)). Consequently, φ is onto. Hence φ is a bijection. Proposition 4.24. Let M be a Γ-ring with unities and L be its left operator ring and A be an IFCI of L. Then A+ is an IFCI of M , where A+ is explained in Definition (4.5). Proof. By Proposition (4.6), A+ is an IFI of Γ-ring M . Let x ∈ M and f ∈ ′ Aut(M ). Then by Proposition (4.18), f + ∈ Aut(L). Hence by using Definition (4.5) and (4.17) we obtain µ(A+ )f (x) = µA+ (f (x)) = Infα∈Γ (µA ([f (x), α])) = Infα∈Γ (µA (f + ([x, α]))) = Infα∈Γ (µA ([x, α])) = µA+ (x). Similarly, we can prove ν(A+ )f (x) = νA+ (x), i.e., (A+ )f (x) = A+ (x), ∀f ∈ Aut(M ). Hence A+ is an IFCI of M . Proposition 4.25. Let M be a Γ-ring with unities and L be its left operator ring ′ ′ and B be an IFCI of M . Then B + is an IFCI of L, where B + is explained in Definition (4.5). ′ P Proof. By Proposition (4.7), B + is an IFI of L. Let i [xi , αi ] ∈ L and g ∈ ′ Aut(L). Then by Theorem (4.23) ∃, f ∈ Aut(M ) such that f + = g. Now µ(B +′ )g ( X [xi , αi ]) = µB +′ (g( X ′ [xi , αi ])) = µB +′ (f + ( = µ B +′ ( X [xi , αi ])) i i i X [f (xi ), αi ]) = Infm∈M (µB ( i X f (xi )αi m)) i = Infn∈M (µB ( X f (xi )αi f (n)))[ As f is a bijection so f (n) = m ] i = Infn∈M (µB ( X f (xi αi n))) = Infn∈M (µB ( i = µ B +′ ( X X xi αi n)) i [ As B is IFCI of M ] [xi , αi ]). i P P Similarly, we can prove ν(B +′ )g ( i [xi , αi ]) = νB +′ ( i [xi , αi ]), i.e., ′ ′ P ′ P (B + )g ( i [xi , αi ]) = B + ( i [xi , αi ]), ∀g ∈ Aut(L). Hence B + is an IFCI of L. Intuitionistic Fuzzy Characteristic Ideal of a Γ-ring 67 Theorem 4.26. Let M be a Γ-ring with unities and L be its left operator ring. Then ∃ a one to one map between the set of all IFCIs of M and the set of all IFCIs of L. Proof. Let φ be a mapping from the set of all IFCIs of M to that of L. let D be ′ an IFCI of M . Let us define φ(D) = D+ . Then by Proposition (4.25), φ(D) is an IFCI of L. Let A be an IFCI of L. Then by Proposition (4.24), A+ is an IFCI ′ of M . Then by Theorem (4.10), (A+ )+ = A, i.e., φ(A+ ) = A. Thus φ is onto. ′ ′ ′ Again if for D1 , D2 of M such that φ(D1 ) = φ(D2 ) then D1+ = D2+ ⇒ (D1+ )+ = ′ (D2+ )+ ⇒ D1 = D2 (by Theorem (4,10)). Therefore φ is one to one, hence the proof. P P Proposition 4.27. Let M be a Γ-ring with left unity i [ei , δi ], right unity i [γi , ai ] and L be its left operator ring. Let K be a characteristic ideal of L. Then K + is a characteristic ideal of M . ′ Proof. Let f ∈ Aut(M ). Then by Proposition (4.18), f + ∈ Aut(L). Hence ′ f + (K) = K. Let f (x) ∈ f (K +) ), where x ∈ K + . Then [x, α] ∈ K, ∀α ∈ Γ. ′ ′ Hence f + ([x, α]) ∈ f + (K), ∀α ∈ Γ ⇒ [f (x), α] ∈ K, ∀α ∈ Γ ⇒ f (x) ∈ K + . Thus f (K + ) ⊆ K + . Hence f −1 (K + ) ⊆ K + (since f ∈ Aut(M ) ⇒ f −1 ∈ Aut(M ) ⇒ K + ⊆ f (K + ). Hence f (K + ) = K + . Consequently, K + is a characteristic ideal of M. Theorem 4.28. Let M be a Γ-ring with unities and L be its left operator ring. Then ∃ an inclusion preserving one to one between the set of all characteristic ide′ als of M and the set of all characteristic ideals of L via the mapping K → K + . ′ Proof. Let us denote the mapping ψ : K → K + . Let K, I be two characteristic ′ ′ ′ ′ ideals of M such that ψ(K) = ψ(I). Then K + = I + ⇒ (K + )+ = (I + )+ ⇒ K = I. (by Theorem (4.15). So ψ is one-one. Let K be a characteristic ideal of L, then by proposition (4.27) K + is a char′ ′ ′ acteristic ideal of M . Also (K + )+ = K. Thus ψ(K + ) = (K + )+ = K. Hence ψ is onto. From Theorem (4.15), it follows that ψ is inclusion preserving. 5. Conclusion In this paper, we studied the notion of intuitionistic fuzzy characteristic ideal of a Γ-ring. We have constructed an example of an intuitionistic fuzzy ideal which is not an intuitionistic fuzzy characteristic ideal. A connection between the intuitionistic fuzzy characteristic ideal with its level cut sets has been studied. The relationships between the set of all automorphisms of Γ-ring and the corresponding automorphisms of its operator rings have been investigated. We proved that there 68 South East Asian J. of Mathematics and Mathematical Sciences exists a one to one map between the set of all intuitionistic fuzzy characteristic ideals of Γ-ring and the set of all intuitionistic fuzzy characteristic ideals of its operator ring. We see that these structures are useful in developing the concepts like intuitionistic fuzzy prime ideals, intuitionistic fuzzy primary ideals and intuitionistic fuzzy semiprime ideals of a Γ-ring. Acknowledgements We thank the anonymous reviewer(s) for the constructive and insightful comments, which have helped us to substantially improve our manuscript. The second author takes this opportunity to express her gratitude to Lovely Professional University, Phagwara, for giving the platform for the research work to be conducted. References [1] Aggarwal, A. K., Mishra, P. K., Verma, S., Sexena, R., A study of some theorems on fuzzy prime ideals of Γ-rings, Available on SSRN-Elsevier, (2019), 809-814. [2] Alhaleem, N. A., Ahmad, A. G., Intuitionistic fuzzy normed subrings and intuitionistic fuzzy normed ideals, Mathematics, 8(9) (2020), 1594. [3] Atanassov, K. T., Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 87-96. [4] Atanassov, K. 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