Notes on Intuitionistic Fuzzy Sets Print ISSN 1310–4926, Online ISSN 2367–8283 2022, Volume 28, Number 1, Pages 11–22 DOI: 10.7546/nifs.2022.28.1.11-22 On the translational invariant intuitionistic fuzzy subset of a Γ-ring Hem Lata1 and P. K. Sharma2 1 2 Research Scholar, Lovely Professional University Phagwara, Punjab, India e-mail: goyalhema1986@gmail.com Post-Graduate Department of Mathematics, D.A.V. College Jalandhar, Punjab, India e-mail: pksharma@davjalandhar.com Received: 10 December 2021 Revised: 18 March 2022 Accepted: 20 March 2022 Abstract: In this paper, we introduce the notion of translational invariant intuitionistic fuzzy subset of a Γ-ring and generalize some notions of a ring to a Γ-ring. Also, we define ideals of a Γ-ring generated by an intuitionistic fuzzy subset with an element of Γ-ring and study their properties. The notion of units, associate, prime element, irreducible element are also generalized with respect to the intuitionistic fuzzy subset of a Γ-ring. Further, we study the properties of homomorphic image and pre-image of translational invariant intuitionistic fuzzy subset under the Γ-ring homomorphism and we prove that every homomorphic image of a prime ideal of a Γ-ring generated by an Aγ -prime element and translational invariant and f -invariant intuitionistic fuzzy subset is also a prime ideal. Keywords: Γ-Ring, Translational invariant intuitionistic fuzzy subset (TIIFS), f -invariant intuitionistic fuzzy subset, Aγ -unit, Aγ -prime element. 2020 Mathematics Subject Classification: 16Y99, 03F55, 03G25. 1 Introduction The notion of a Γ-ring was introduced by N. Nobusawa [9] as more general than the notion of a ring. W. E. Barnes [2] weakened slightly the conditions in the definition of Γ-rings in the sense of N. Nobusawa. The structure of Γ-rings can be found in [11]. The notion of intuitionistic 11 fuzzy set was introduced by K. T. Atanassov [1] to generalize the notion of fuzzy set given by L. A. Zadeh [14]. R. Biswas [4] was the first one to introduce the concept of intuitionistic fuzzy subgroup of a group and established many important properties. The notion of intuitionistic fuzzy subring and ideal in a ring was introduced by K. Hur et al. in [6, 7]. K. H. Kim et al. in [8] have studied intuitionistic fuzzy ideal of Γ-rings which was further studied by N. Palaniappan et al. in [10]. A. K. Ray [12] introduced the concept of translational invariant fuzzy subset in a ring. A. K. Ray and T. Ali in [13] also studied ideals and divisibility in a ring with respect to a fuzzy subset. Y. Bhargavi [3] studied the translational invariant vague set of a Γ-semiring. The purpose of this paper is to generalize some of the classical results of ring theory using the notion of a translational invariant intuitionistic fuzzy subset (TIIFS) of a Γ-ring. 2 Preliminaries In this section, we list some basic concepts and definitions on Γ-rings theory and intuitionistic fuzzy sets theory, which are necessary for the better understanding of the paper. Definition 2.1 ([2]). If (M, +) and (Γ, +) are additive Abelian groups, then M is called a Γ-ring if there exists a mapping f : M ×Γ×M → M , where f (x, α, y) is denoted by xαy, x, y ∈ M, γ ∈ Γ satisfying the following conditions: (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 γ ∈ Γ. These conditions are further strengthened by defining another function g : Γ × M × Γ → Γ, where g(α, x, β) is denoted by αxβ, x ∈ M , α, β ∈ Γ, satisfying the following conditions for all x, y, z ∈ M and for all α, β, γ ∈ Γ, ′ (1 ) xαy ∈ M , αxβ ∈ Γ. ′ (2 ) (x + y)αz = xαz + yαz, x(α + β)y = xαy + xβy, xα(y + z) = xαy + xαz. ′ (3 ) (xαy)βz = xα(yβz). ′ (4 ) xαy = 0M for all x, y ∈ M implies α = 0Γ . We then have a Γ-ring in the sense of Nobusawa [9]. Definition 2.2 ([2, 10]). A subset N of a Γ-ring M is a left (right) ideal of M if N is an additive subgroup of M and M ΓN = {xαy|x ∈ M, α ∈ Γ, y ∈ N }, (N ΓM ) is contained in N . If N is both a left and a right ideal, then N is a two-sided ideal, or simply an ideal of M . 12 Definition 2.3 ([13]). A Γ-ring M is said to be a commutative Γ-ring if xγy = yγx, ∀x, y ∈ M, γ ∈ Γ. Definition 2.4 ([13]). Let M be a Γ-ring. An element e ∈ M is said to be unity if for each x ∈ M there exists γ ∈ Γ such that xγe = eγx = x. Definition 2.5 ([13]). An ideal P of a Γ-ring M is said to be prime ideal of M if for any x, y ∈ M, γ ∈ Γ, xγy ∈ P implies that x ∈ P or y ∈ P . ′ ′ Definition 2.6 ( [2, 13]). Let M and M be two Γ-rings. Then f : M → M is called a Γ-homomorphism if • f (x + y) = f (x) + f (y) • f (xγy) = f (x)γf (y), for all x, y ∈ M, γ ∈ Γ. Definition 2.7 ([1]). An intuitionistic fuzzy set A in X can be represented as an object of the form A = {hx, µA (x), νA (x)i : 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. Remark 2.8 ([1]). (i) When µA (x) + νA (x) = 1, i.e., νA (x) = 1 − µA (x) = µAc (x), then A is called a fuzzy set. (ii) An intuitionistic fuzzy set (IFS) A = {hx, µA (x), νA (x)i : x ∈ X} is shortly denoted by A(x) = (µA (x), νA (x)), for all x ∈ X. Proposition 2.9 ([1]). If A, B be 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. 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 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 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 all x ∈ X. Also the IFS A of X is said to be f -invariant if for any x, y ∈ X, whenever f (x) = f (y) implies A(x) = A(y). 3 Translational invariant intuitionistic fuzzy subset of a Γ-ring Throughout this section, M is a Γ-ring with unities and the zero element θ. Definition 3.1. Let A be an intuitionistic fuzzy subset of M . A is called a left translational invariant intuitionistic fuzzy subset with respect to the internal addition if A(x) = A(y) implies that A(x + m) = A(y + m), for all x, y, m ∈ M . Again A is called a left translational invariant 13 intuitionistic fuzzy subset with respect to the external multiplication if A(x) = A(y) implies that A(mγx) = A(mγy), for all x, y, m ∈ M and for all γ ∈ Γ. Similarly, we can define the notion of right translational invariant intuitionistic fuzzy subset with respect to the operation (addition, multiplication) in M . Remark 3.2. An IFS A is said to be commutative under internal addition (or external multiplication) on M if A(x + y) = A(y + x) (or A(xγy) = A(yγx)), for all x, y ∈ M, γ ∈ Γ. Therefore, when A is commutative, then the two notion coincides. In this case, we say that A is a translational invariant intuitionistic fuzzy subset (TIIFS) of M with respect to the operation + (or ×). Example 3.3. Consider the Γ-ring M , where M = Z the ring of integers and Γ = 2Z, the ring of even integers and xγy denotes the usual product of integers x, γ, y. Let A = (µA , νA ) be an intuitionistic fuzzy subset of M defined by   1, 0, if x is an even integer if x is an even integer µA (x) = ; νA (x) = 0.5, if x is an odd integer 0.3, if x is an odd integer. Then it is easy to verify that A is an TIIFS of M with respect to both the operation + and ×. Example 3.4. Consider the Γ-ring M , where M = {[aij ] : aij ∈ Z2 , i = 1, j = 1, 2}, the set of (1 × 2) matrices whose entries are from Z2 and Γ = {[aij ] : aij ∈ Z2 , i = 1, 2, j = 1}, the set of (2 × 1) matrices whose entries are from Z2 . Let A = (µA , νA ) be an intuitionistic fuzzy subset of R defined by     0.2, if a11 = a12 = 0 0.7, if a = a = 0   11 12       0.2, if a = 1, a = 0 0.7, if a = 1, a = 0 11 12 11 12 ; νA (aij ) = µA (aij ) = 0.5, if a11 = 0, a12 = 1 0.3, if a11 = 0, a12 = 1         0.5, if a = a = 1. 0.3, if a = a = 1. 11 12 11 12 Then it is easy to verify that A is an TIIFS of M with respect to both the operation addition of matrices and multiplication of matrices defined on M . From this point onwards, every intuitionistic fuzzy subset A of a Γ-ring M satisfies the property A(−x) = A(x), for all x ∈ M . Proposition 3.5. Let A be a TIIFS with respect to both internal addition and external multiplication operations defined on M . Then for any m ∈ M the set L(m, γ, A) = {x : x ∈ M such that A(x) = A(yγm), for some y ∈ M } is a left ideal of M . Proof. Clearly L(m, γ, A) 6= ∅, since θ ∈ L(m, γ, A) as A(θ) = A(θγm). Let x1 , x2 ∈ L(m, γ, A). Then A(x1 ) = A(y1 γm) and A(x2 ) = A(y2 γm), for some y1 , y2 ∈ M . Now A(x1 ) = A(y1 γm) ⇒ A(x1 − x2 ) = A(y1 γm − x2 ) = A(x2 − y1 γm) (i) A(x2 ) = A(y2 γm) ⇒ A(x2 − x1 ) = A(y2 γm − x1 ) = A(x1 − y2 γm) (ii) and 14 From (i) and (ii) we get A(x1 −y2 γm) = A(x2 −y1 γm) ⇒ A(x1 −x2 ) = A(y1 γm−y2 γm) = A((y1 − y2 )γm). Thus, x1 − x2 ∈ L(m, γ, A), since (y1 − y2 ) ∈ M . Also, for any y3 ∈ M and γ1 ∈ Γ, we have A(y3 γ1 x1 ) = A(y3 γ1 (y1 γm)) = A((y3 γ1 y1 )γm) ⇒ y3 γ1 x1 ∈ L(m, γ, A) for any y3 ∈ M and for any γ1 ∈ Γ. Hence L(m, γ, A) is a left ideal of M . Analogously we can prove: Proposition 3.6. Let A be a TIIFS with respect to both internal addition and external multiplication operations defined on M . Then for any m ∈ M the set R(m, γ, A) = {x : x ∈ M such that A(x) = A(mγy), for some y ∈ M } is a right ideal of M . Remark 3.7. If M is a commutative Γ-ring, then L(m, γ, A) = R(m, γ, A), ∀m ∈ M and for all γ ∈ Γ. Remark 3.8. We observe that for any m ∈ M and γ ∈ Γ, the ideal M γm = {xγm : x ∈ M } of M is contained in the left ideal L(m, γ, A). Also for any m ∈ M and γ ∈ Γ, the ideal mγM = {mγx : x ∈ M } of M is contained in the right ideal R(m, γ, A). Definition 3.9. L(m, γ, A) is called left A-principal ideal of M generated by m, γ and A, and R(m, γ, A) is called right A-principal ideal of M generated by m, γ and A. Definition 3.10. If L(m, γ, A) = R(m, γ, A) for all m ∈ M and γ ∈ Γ, then the ideal is denoted by I(m, γ, A) and is called A-principal ideal of m generated by m, γ and A. Definition 3.11. A Γ-ring M is called A-principal ideal Γ-ring if A is commutative and every ideal of M is an A-principal ideal generated by some m ∈ M, γ ∈ Γ and A. Definition 3.12. An element a ∈ M with A(a) 6= A(θ) is called an Aγ -unit of M , where γ ∈ Γ if ′ ′ ′ ′ there exists an element a ∈ M such that A(a ) 6= A(θ) and A(aγa γm) = A(m) = A(a γaγm) for all m ∈ M . From the definition it follows that γ 6= 0Γ . In a Γ-field every element a(6= θ) is an Aγ -unit for all γ(6= 0Γ ) ∈ Γ. Proposition 3.13. If a is an Aγ -unit of M , then L(m, γ, A) = R(m, γ, A) = M , for all γ ∈ Γ. ′ ′ ′ Proof. As a is an Aγ -unit of M, ∃ a ∈ M such that A(a ) 6= A(θ) and A(aγa γm) = A(m) = ′ ′ A(a γaγm), for all m ∈ M . Let x ∈ M . Then A(x) = A(aγa γm) ⇒ x ∈ R(m, γ, A), since ′ a γx ∈ M . Therefore, M ⊆ R(m, γ, A). Similarly, M ⊆ L(m, γ, A). Hence L(m, γ, A) = R(m, γ, A) = M for γ ∈ Γ, γ 6= 0Γ . Proposition 3.14. Let A be a TIIFS with respect to external multiplication defined on M and a, b ∈ M . Then, a ∈ L(b, γ, A) for some γ ∈ Γ ⇒ L(a, γ, A) ⊆ L(b, γ, A) and a ∈ R(b, γ, A) for some γ ∈ Γ ⇒ R(a, γ, A) ⊆ R(b, γ, A). 15 Proof. Let a ∈ L(b, γ, A), then A(a) = A(xγb), for some x ∈ M . Let m ∈ L(a, γ, A). Then A(m) = A(yγa) for some y ∈ M . Now A(a) = A(xγb) ⇒ A(yγa) = A(yγxγb) ⇒ A(m) = A(yγxγb) ⇒ m ∈ L(b, γ, A). Hence L(a, γ, A) ⊆ L(b, γ, A). Similarly, we can prove R(a, γ, A) ⊆ R(b, γ, A). Remark 3.15. We observe that L(a, γ, A) = {m ∈ M : A(m) = A(θ)} = MA , for any γ ∈ Γ. Proposition 3.16. Let A be a TIIFS with respect to the external multiplication defined on M and a, b ∈ M . Then A(a) = A(b) ⇒ L(a, γ, A) = L(b, γ, A); R(a, γ, A) = R(b, γ, A). Proof. Let A(a) = A(b). Suppose m ∈ L(a, γ, A). Then A(m) = A(xγa) for some x ∈ M . Now A(a) = A(b) implies A(xγa) = A(xγb). Hence, A(m) = A(xγb, so m ∈ L(a, γ, A). Thus, L(a, γ, A) ⊆ L(b, γ, A). Similarly, we can show that L(b, γ, A) ⊆ L(a, γ, A). Consequently, L(a, γ, A) = L(b, γ, A). In a similar way, we can prove R(a, γ, A) = R(b, γ, A). In the next two sections, M is assumed to be a commutative Γ-ring with right and left unities and A is assumed to be a translational invariant intuitionistic fuzzy subset of M with respect to both internal addition and external multiplication defined on M satisfying A(x) = A(−x), ∀x ∈ M . Henceforth, the ideal generated by an element a ∈ M , γ ∈ Γ with respect to A will be denoted by I(a, γ, A) and it will be assumed that a ∈ I(a, γ, A) for all γ ∈ Γ. 4 A-divisors of zero, A-associates Definition 4.1. An element a ∈ M with A(a) 6= A(θ) is said to be an Aγ -divisor of zero for γ ∈ Γ, γ 6= 0Γ if there exists some b ∈ M with A(b) 6= A(θ) such that A(aγb) = A(θ). Henceforth, we shall assume that M contains no Aγ -divisor of zero. Definition 4.2. Let a, b ∈ M and A(a) 6= A(θ). We say that a divides b with respect to A and γ ∈ Γ or a is an Aγ - divisor of b, written as (a/b)Aγ , if there exists c ∈ M such that A(b) = A(aγc). Theorem 4.3. Let a, b ∈ M be such that A(a) 6= A(b) and A(a) 6= A(θ). Then (a/b)Aγ if and only if I(b, γ, A) ⊆ I(a, γ, A), for γ ∈ Γ. Proof. Suppose that (a/b)Aγ . Then A(b) = A(cγa) for some c ∈ M , which implies that b ∈ I(a, γ, A) and, therefore, I(b, γ, A) ⊆ I(a, γ, A). Conversely, let I(b, γ, A) ⊆ I(a, γ, A). As b ∈ I(b, γ, A) ⊆ I(a, γ, A) hence, A(b) = A(cγa), for some c ∈ M . Also, A(a) 6= A(θ). Hence (a/b)Aγ . Definition 4.4. Let a, b ∈ M \MA be such that A(a) 6= A(b). We say that a and b are A-associates with respect to γ ∈ Γ if (a/b)Aγ and (b/a)Aγ . Proposition 4.5. Let a, b ∈ M \MA . Then a, b are A-associates with respect to γ ∈ Γ if and only if A(a) = A(bγu) for some Aγ -unit u ∈ M . 16 Proof. Let a, b be A-associates with respect to γ. Then (a/b)Aγ and (b/a)Aγ . So A(b) = A(aγd) and A(a) = A(bγc) for some c, d ∈ M . Hence A(a) = A(bγc) = A(aγdγc) ⇒ A(aγx) = A(aγdγcγx) ⇒ A(aγx − aγdγcγx) = A(θ) ⇒ A(aγ(x − dγcγx)) = A(θ) ⇒ A(x − dγcγx) = A(θ); since A(a) 6= A(θ) and R is without Aγ -divisor of zero. ⇒ A(x) = A(dγcγx), for all x ∈ M ⇒ c and d are Aγ -units in M . Hence A(a) = A(bγc), where c is an Aγ -unit in M . Conversely, suppose that A(a) = A(bγu), for some Aγ -unit u in M . Now, A(a) = A(bγu) ⇒ (b/a)Aγ . Since u is an Aγ -unit, there exists v ∈ M \MA such that A(uγvγx) = A(x), for all x ∈ M . Hence A(a) = A(bγu) ⇒ A(aγv) = A(bγuγv) = A(b). This shows that (a/b)Aγ . Thus we find (a/b)Aγ and (b/a)Aγ . Hence a, b are Aγ -associates. Corollary 4.6. Let a, b ∈ M \MA . If a, b are Aγ -associates, then I(a, γ, A) = I(b, γ, A). Proof. Suppose that a and b are Aγ -associates. Then by Proposition 4.5, A(a) = A(uγb), for some Aγ -unit u ∈ M . Then, a ∈ I(b, γ, A), and so I(a, γ, A) ⊆ I(b, γ, A). Since u is an Aγ -unit of M , there exists v ∈ M \MA such that A(uγvγx) = A(x), for all x ∈ M . Hence A(bγuγv) = A(b). Thus A(b) = A(aγv), and so b ∈ I(a, γ, A). Therefore I(b, γ, A) ⊆ I(a, γ, A). Consequently, I(a, γ, A) = I(b, γ, A). Definition 4.7. Suppose a ∈ M \MA and a is not an Aγ -unit for γ ∈ Γ. Then a is said to be Aγ -irreducible if A(a) 6= A(b), A(a) 6= A(c) and A(a) = A(bγc) implies either b or c is an Aγ -unit, where b, c ∈ M . Definition 4.8. Suppose a ∈ M \MA and a not an Aγ -unit for γ ∈ Γ. Then a is said to be Aγ -prime if A(a) 6= A(b), A(a) 6= A(c) and (a/bγ1 c)Aγ implies (a/b)Aγ or (a/c)Aγ , where γ ∈ Γ. Proposition 4.9. In the Γ-ring M with no Aγ -divisors of zero, any Aγ -prime is Aγ -irreducible. Proof. Let a be Aγ -prime. Suppose A(a) 6= A(b), A(a) 6= A(c) and A(a) = A(bγc) for γ ∈ Γ, b, c ∈ M . We can say that (a/bγc)Aγ . Since a is Aγ -prime, either (a/b)A or (a/c)A . Suppose (a/b)Aγ . As A(a) 6= A(b), A(b) = A(aγd) for some d ∈ M . Now A(a) = A(bγc) = A(aγdγc) ⇒ A(aγx) = A(aγdγcγx), for x ∈ M ⇒ A(aγx − aγdγcγx) = A(θ) ⇒ A(aγ(x − dγcγx)) = A(θ) ⇒ A(x − dγcγx) = A(θ), since A(a) 6= A(θ) and M is without Aγ -divisor of zero. ⇒ A(x) = A(dγcγx), for all x ⇒ c is a Aγ -unit. Similarly, if (a/c)Aγ then we can show that b is an Aγ -unit. Hence a is Aγ -irreducible. Theorem 4.10. Suppose that a ∈ M \MA and a is not an Aγ -unit. Then a is Aγ -irreducible if and only if the ideal I(a, γ, A) is maximal among all ideals I(b, γ, A), where b ∈ M and A(a) 6= A(b). 17 Proof. (i) Suppose that a is Aγ -irreducible. Let I(a, γ, A) ⊆ I(b, γ, A) 6= M for some b ∈ M with A(b) 6= A(a). Now a ∈ I(a, γ, A) ⊆ I(b, γ, A) and so A(a) = A(cγb) for some c ∈ M \MA . Now, if A(a) = A(c), then A(c) = A(cγb), which implies A(cγx) = A(cγbγz), for all x ∈ M . Now M is without Aγ -divisor of zero and A(c) 6= A(θ), so A(x) = A(bγx) for all x ∈ M . Hence I(b, γ, A) = M , which is not the case. Hence A(a) 6= A(c). As a is Aγ -irreducible, so either b is an Aγ -unit or c is an Aγ -unit. Since I(b, γ, A) 6= M so by Proposition 4.9, we find that b is not an Aγ -unit. So there exists u ∈ M \MA such that A(cγuγx) = A(uγcγx) = A(x), for all x ∈ M . Thus A(b) = A(cγuγb). Again, A(a) = A(bγc) implies A(aγu) = A(b). Hence b ∈ I(a, γ, A) and so I(b, γ, A) ⊆ I(a, γ, A). Consequently, I(b, γ, A) = I(a, γ, A). Thus I(a, γ, A) is maximal. Conversely, assume that I(a, γ, A) is maximal. Assume that A(a) = A(cγd) where c, d ∈ M and A(a) 6= A(c), A(a) 6= A(d). Then a ∈ I(d, γ, A) and so I(a, γ, A) ⊆ I(d, γ, A). Hence by our hypothesis either I(a, γ, A) = I(d, γ, A), or I(d, γ, A) = M . If I(a, γ, A) = I(d, γ, A), then d ∈ I(d, γ, A) = I(a, γ, A). Therefore, A(d) = A(mγa), for some m ∈ M . This gives A(cγd) = A(cγmγa). Thus we have A(a) = A(cγmγa) and so A(aγ(x − cγmγx)) = A(θ), for all x ∈ M . Since M is without Aγ -divisors of zero and A(a) 6= A(θ), we have A(x) = A(cγmγx), for all x ∈ M . This shows that c is an Aγ -unit. If I(d, γ, A) = M , then A(d) = A(dγm), for some m ∈ M . Again A(m) = A(dγy), for some y ∈ M . Therefore A(d) = A(dγm) = A(dγdγy). Hence, A(dγx) = A(dγdγyγx). Thus A(x) = A(dγyγx). This shows that d is an Aγ -unit. Theorem 4.11. Suppose that a ∈ M \MA and a is not an Aγ -unit. Then a is Aγ -prime if and only if for x, y ∈ M, γ1 ∈ Γ, xγ1 y ∈ I(a, γ, A) implies either that x ∈ I(a, γ, A) or y ∈ I(a, γ, A), where A(a) 6= A(x), A(a) 6= A(y). Proof. Suppose that a is Aγ -prime and x, y ∈ M , γ1 ∈ Γ, xγ1 y ∈ I(a, γ, A) implies either x ∈ I(a, γ, A) or y ∈ I(a, γ, A), where A(a) 6= A(x), A(a) 6= A(y). Then A(xγ1 y) = A(aγm) for some m ∈ M , which shows that (a/xγ1 y)Aγ . As a is Aγ -prime, so either (a/x)Aγ or (a/y)Aγ . If (a/x)Aγ , then there exists x1 ∈ M such that A(x) = A(aγx1 ) which implies x ∈ I(a, γ, A). Similarly, if (a/y)Aγ , then there exists y1 ∈ M such that A(y) = A(aγy1 ), which implies y ∈ I(a, γ, A). Conversely, let for x, y ∈ M, γ1 ∈ Γ, xγ1 y ∈ I(a, γ, A) implies either x ∈ I(a, γ, A) or y ∈ I(a, γ, A), where A(a) 6= A(x), A(a) 6= A(y). We have to prove a is Aγ -prime. Let (a/xγ1 y)Aγ , where x, y ∈ M, γ1 ∈ Γ, A(a) 6= A(x), A(a) 6= A(y). A(xγ1 y) = A(aγm), for some m ∈ M . Thus xγ1 y ∈ I(a, γ, A). Now from given condition either x ∈ I(a, γ, A), or y ∈ I(a, γ, A). If x ∈ I(a, γ, A), then A(x) = A(aγ1 m1 ) for some m1 ∈ M . Thus (a/x)Aγ . If y ∈ I(a, γ, A), then A(y) = A(aγ1 m2 ) for some m2 ∈ M . Thus (a/y)Aγ . This proves that a is Aγ -prime. 18 5 Images and inverse images under Γ-ring homomorphisms In this section, we discuss the invariance of translational invariance property of an intuitionistic fuzzy subset under Γ-ring homomorphism. We also study the algebraic nature of ideals under Γ-ring homomorphism. ′ ′ Proposition 5.1. Let M and M be Γ-rings and f be a Γ-homomorphism from M into M . Let ′ B be a TIIFS of M . Then f −1 (B) is a TIIFS of M . Proof. Let a, b ∈ M and f −1 (B)(a) = f −1 (B)(b). Then B(f (a)) = B(f (b)). Let x ∈ M and ′ ′ f (x) = y ∈ M . Since B is a TIIFS of M and B(f (a)) = B(f (b)), we have B(f (a) + y) = B(f (b) + y) and B(f (a)γy) = B(f (b)γy), B(yγf (a)) = B(yγf (b)). Now B(f (a) + y) = B(f (b) + y) implies B(f (a) + f (x)) = B(f (b) + f (x)), and so B(f (a + x)) = B(f (b + x)). Hence f −1 (B)(a + x) = f −1 (B)(b + x). On the other hand, from B(f (a)γy) = B(f (b)γy) and B(yγf (a)) = B(yγf (b)), we get B(f (a)γf (x)) = B(f (b)γf (x)) and B(f (x)γf (a)) = B(f (x)γf (b)), and so B(f (aγx)) = B(f (bγx)) and B(f (xγa)) = B(f (xγb)). Thus, we have f −1 (B)(aγx) = f −1 (B)(bγx) and f −1 (B)(xγa) = f −1 (B)(xγb)∀a, b, x ∈ M and γ ∈ Γ. Consequently, f −1 (B) is TIIFS of M . ′ ′ Proposition 5.2. Let M and M be Γ-rings and f be a Γ-homomorphism from M onto M . Let ′ A be a TIIFS of M . If A is f -invariant, then f (A) is a TIIFS of M . Proof. Suppose that A is f -invariant. Then ∀x, y ∈ M , f (x) = f (y) implies A(x) = A(y). ′ As f is onto, for any a ∈ M , µf (A) (a) = sup{µA (x) : x ∈ M, f (x) = a} and νf (A) (a) = inf{νA (x) : x ∈ M, f (x) = a}. Let x, y ∈ M and f (x) = a = f (y). Then f (x) = f (y), and so A(x) = A(y). So µf (A) (a) = µA (x) and νf (A) (a) = νA (x). Hence f (A)(a) = A(x), where ′ x ∈ M and f (x) = a. Thus ∀a ∈ M , f (A)(a) = A(x), where x ∈ M and f (x) = a. Now, let ′ a, b ∈ M , and f (A)(a) = f (A)(b). Then A(x) = A(y), where x, y ∈ M , and f (x) = a, f (y) = ′ b. Let c ∈ M be such that f (z) = c, where z ∈ M . Then, a + c = f (x) + f (z) = f (x + z) and b + c = f (y) + f (z) = f (y + z). Hence f (A)(a + c) = A(x + z) and f (A)(b + c) = A(y + z). Again, for γ ∈ Γ, aγc = f (x)γf (z) = f (xγz), cγa = f (z)γf (x) = f (zγx), bγc = f (y)γf (z) = f (yγz), and cγb = f (z)γf (y) = f (zγy). Since A is translational invariant, A(x + z) = A(y + z), A(xγz) = A(yγz), and A(zγx) = A(zγy). Hence, f (A)(a + c) = ′ f (A)(b + c), f (A)(aγc) = f (A)(bγc), and f (A)(cγa) = f (A)(cγb) for all c ∈ M . Hence, f (A) ′ is a TIIFS of M . ′ ′ Theorem 5.3. Let M and M be Γ-rings and f be a Γ-homomorphism from M onto M and A be a TIIFS of M . If A is f -invariant then, f (I(a, γ, A)) = I(f (a), γ, f (A)), ∀a ∈ M, γ ∈ Γ. Proof. Suppose that A is f -invariant. Let y ∈ I(f (a), γ, f (A)). Then f (A)(y) = f (A)(sγf (a)) ′ ′ for some s ∈ M . Since y, s ∈ M and f is onto, there exist x, r ∈ M such that f (x) = y and f (r) = s. Thus f (A)f (x) = f (A)(f (r)γf (a)) = f (A)(f (rγa)). Since A is translational invariant, by what we have proved in Proposition 5.2, we get f (A)(f (x)) = A(x) and 19 f (A)(f (rγa)) = A(rγa). Thus A(x) = A(rγa), which implies x ∈ I(a, γ, A), and so f (x) ∈ f (I(a, γ, A)), i.e., y ∈ f (I(a, γ, A)). Consequently, I(f (a), γ, f (A)) ⊆ f (I(a, γ, A)). Again, let y ∈ f (I(a, γ, A)). Then there exists x ∈ I(a, γ, A) such that f (x) = y. Also, x ∈ I(a, γ, A) implies A(x) = A(aγr) for some r ∈ M . Now, µf (A) (y) = Sup{µA (x) : f (x) = y} = µA (x), since A is f -invariant = µA (aγr). Similarly, we can prove νf (A) (y) = νA (aγr). Hence f (A)(y) = A(aγr). Also, if f (r) = s, we have f (A)(f (a)γs) = f (A)(f (a)γf (r)) = f (A)(f (aγr)) = A(aγr), since A is f -invariant so f −1 f (aγr) = aγr. Thus f (A)(y) = f (A)(f (a)γs) which implies y ∈ I(f (a), γ, f (A)). Hence f (I(a, γ, A)) ⊆ I(f (a), γ, f (A)), a ∈ M . Consequently, f (I(a, γ, A)) = I(f (a), γ, f (A)), for a ∈ M, γ ∈ Γ. ′ ′ Proposition 5.4. Let M and M be Γ-rings and f be a Γ-homomorphism from M onto M . Let ′ ′ ′ ′ B be a TIIFS of M . Let a ∈ M . Then ∀a, b ∈ f −1 (a ), I(a, γ, f −1 (B)) = I(b, γ, f −1 (B)); ′ provided that f −1 (a ) contains more than one element. Proof. Let x ∈ I(a, γ, f −1 (B)). Then f −1 (B)(x) = f −1 (B)(rγa) for some r ∈ M and ′ so f −1 (B)(x) = B(f (rγa)). Thus f −1 (B)(x) = B(f (a)γf (r)). Since a, b ∈ f −1 (a ), ′ f (a) = f (b) = a and hence we have f −1 (B)(x) = B(f (b)γf (r)) = B(f (bγr)) = f −1 (B)(bγr). This shows that x ∈ I(b, γ, f −1 (B)). Hence I(a, γ, f −1 (B)) ⊆ I(b, γ, f −1 (B)). ′ ′ Now let y ∈ I(b, γ, f −1 (B)). Then f −1 (B)(y) = f −1 (B)(bγr ) for some r ∈ M , and ′ ′ ′ ′ so f −1 (B)(y) = B(f (bγr )) = B(f (b)γf (r )), Since a, b ∈ f −1 (a ), f (a) = a = f (b) ′ ′ ′ and hence we have f −1 (B)(y) = B(f (a)γf (r )) = B(f (aγr )) = f −1 (B)(aγr ). This shows that y ∈ I(a, γ, f −1 (B)). Hence, I(b, γ, f −1 (B)) ⊆ I(a, γ, f −1 (B)). Consequently, ′ I(a, γ, f −1 (B)) = I(b, γ, f −1 (B)) ∀a, b ∈ f −1 (a ). ′ ′ Theorem 5.5. Let M and M be Γ-rings and f be a Γ-isomorphism from M onto M . Let B be ′ a translational invariant intuitionistic fuzzy subset of M . Then ′ I(f −1 (y), γ.f −1 (B)) = f −1 (I(y, γ, B)), ∀y ∈ M , γ ∈ Γ. Proof. Let x ∈ I(f −1 (y), γ, f −1 (B)). Then f −1 (B)(x) = f −1 (B)(f −1 (y)γr) for some r ∈ M. ′ = f −1 (B)(f −1 (y)γf −1 (s)), where s ∈ M such that f (r) = s. ⇒ B(f (x)) = f −1 (B)(f −1 (yγs)), since f is bijective = B(f (f −1 (yγs))) = B(yγs). So, we have f (x) ∈ I(y, γ, B), i.e., x ∈ f −1 (I(y, γ, B)). Hence I(f −1 (y), γ, f −1 (B)) ⊆ ′ f −1 (I(y, γ, B)), ∀y ∈ M . Again, let a ∈ f −1 (I(y, γ, B)) then f (a) ∈ I(y, γ, B)) ⇒ 20 ′ ′ B(f (a)) = B(yγs), for some s ∈ M . Also, y, s ∈ M and f is onto implies that there exist x, r ∈ M such that f (x) = y and f (r) = s. Now, B(f (a)) = B(yγs) ⇒ B(f (a)) = B(f (x)γf (r)) = B(f (xγr)) ⇒ f −1 (B)(a) = f −1 (B)(xγr) = f −1 (B)(f −1 (y)γr) ⇒ a ∈ ′ I(f −1 (y), γ, f −1 (B)). Thus, f −1 (I(y, γ, B)) ⊆ I(f −1 (y), γ, f −1 (B)), ∀y ∈ M . Consequently, ′ I(f −1 (y), γ, f −1 (B)) = f −1 (I(y, γ, B)), ∀y ∈ M , γ ∈ Γ. ′ ′ Theorem 5.6. Let M and M be Γ-rings and f be a Γ-homomorphism from M onto M . If A is f -invariant and TIIFS of M . If p is an Aγ -prime element of M , then,f (p) is a f (A)γ -prime ′ element of M . ′ Proof. Let f be a Γ-homomorphism from M onto M . If A is f -invariant and TIIFS of M . Then ′ by Proposition 5.2 f (A) is a TIIFS of M . Suppose that p is an Aγ -prime element of M . Let ′ (f (p)/xγy)f (A)γ , where x, y ∈ M . Since f is onto, there exists a, b ∈ M such that f (a) = x, f (b) = y. Now (f (p)/xγy)f (A)γ ⇒ ∃c ∈ M such that f (A)(xγy) = f (A)(f (p)γf (c)) ⇒ f (A)(f (a)γf (b)) = f (A)(f (pγc)) ⇒ f (A)(f (aγb)) = f (A)(f (pγc)) ⇒ A(aγb) = A(pγc) and so (p/aγb)Aγ . Since p is an Aγ -prime element of M , we have (p/a)Aγ or (p/b)Aγ ⇒ A(a) = A(pγm) or A(b) = A(pγn), for some m, n ∈ M, ∀γ ∈ Γ ⇒ f (A)(f (a)) = f (A)(f (pγm)) or f (A)(f (b)) = f (A)(f (pγn)) ⇒ f (A)(f (a)) = f (A)(f (p)γf (m)) or f (A)(f (b)) = f (A)(f (p)γf (n)) ′ ⇒ (f (p)/f (a))f (A) or (f (p)/f (b))f (A) . Thus, f (p) is a f (A)-prime element of M . ′ Theorem 5.7. Let f be a homomorphism of a Γ-ring M onto a Γ-ring M . Let A be an f -invariant and TIIFS of M . If p is an Aγ -prime element of M , then the homomorphic image ′ of I(p, γ, A) is a prime ideal of M . ′ Proof. Let f be a Γ-homomorphism from M onto M . If A is f -invariant and TIIFS of M , then ′ by Proposition 5.2 f (A) is TIIFS of M . By Theorem 4.11 I(p, γ, A) is a prime ideal of M . By Theorem 5.3 f (I(p, γ, A)) = I(f (p), γ, f (A)). ′ By Theorem 5.6 f (p) is a f (A)γ -prime element of M ′ By Theorem 4.11 f (I(p, γ, A)) is a prime ideal of M . Acknowledgements The first author would like to thank Lovely Professional University Phagwara for providing the opportunity to do research work. 21 References [1] Atanassov, K. T. (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20(1), 87–96. [2] Barnes, W. E. (1966). On the Γ-ring of Nobusawa. Pacific Journal of Mathematics, 18, 411–422. [3] Bhargavi, Y. (2020). A study on translational invariant vague set of a Γ-semiring. Afrika Matematika, DOI: 10.1007/s13370-020-00794-1. [4] Biswas, R. (1989). Intuitionistic fuzzy subgroup. Mathematical Forum, X, 37–46. [5] Burton, D. M. (1970). A first course in rings and ideals. Addison-Wesley, Available online: http://en.b-ok.cc/book/5309376/04beda. [6] Hur, K., Jang, S. Y., & Kang, H. W. (2005). Intuitionistic fuzzy ideals of a ring. Journal of the Korea Society of Mathematical Education, Series B, 12(3), 193–209. [7] Hur, K., Kang, H. W., & Song, H. K. (2003). Intuitionistic fuzzy subgroups and subrings. Honam Mathematical Journal, 25(1), 19–41. [8] Kim, K.H., Jun, Y.B., & Ozturk, M. A. (2001). Intuitionistic fuzzy ideal of Γ-rings. Scientiae Mathematicae Japonicae, 54, 51–60. [9] Nobusawa, N. (1964). On a generalization of the Ring Theory. Osaka Journal of Mathematics, 1, 81–89. [10] Palaniappan, N., Veerappan, P. S., & Ramachandran, M. (2011). Some properties of intuitionistic fuzzy ideal of Γ-rings, Thai Journal of Mathematics, 9(2), 305–318. [11] Ravisanka, T. S., & Shukla, U. S. (1979). Structure of Γ-rings, Pacific Journal of Mathematics, 80(2), 537–559. [12] Ray, A. K. (1999). Quotient group of a group generated by a subgroup and a fuzzy subset. Journal of Fuzzy Mathematics, 7(2), 459–463. [13] Ray, A. K., & Ali, T. (2002). Ideals and divisibility in a ring with respect to a fuzzy subset. Novi Sad Journal of Mathematics, 32(2), 67–75. [14] Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8, 338–353. 22