International Journal of Analysis and Applications Volume 19, Number 3 (2021), 341-359 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-19-2021-341 INTUITIONISTIC FUZZY NORMAL SUBRINGS OVER NORMED RINGS NOUR ABED ALHALEEM∗, ABD GHAFUR AHMAD Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi 43600, Selangor, Malaysia ∗Corresponding author: noorb@ymail.com Abstract. This paper presents the notion of intuitionistic fuzzy normed normal subrings. We investigate the concept of intuitionistic fuzzy normal subrings over normed rings and characterize relevant properties of such subrings. Further, we define direct product of fuzzy normal subrings over normed rings and investigate some related fundamental properties. 1. Introduction After an introduction of fuzzy sets by Zadeh [1] several researchers investigated on the generalization of the concept of fuzzy set. In 1971, Rosenfeld [2] initiated the studies of fuzzy group theory by introducing the concepts of fuzzy subgroupoid and fuzzy subgroup. Later in 1981 [3], Wu introduced the notion of fuzzy normal subgroups of an ordinary group and Liu [4] defined a fuzzy invariant (normal) subgroup, Liu also extended the notion of a subring of a ring and the product of complexes to the fuzzy setting in the same paper. In 1984 [5], Mukharjee and Bhattacharya introduced the concept of fuzzy cosets and studied their relation with normal fuzzy subgroups, proved that for a group G a fuzzy subgroup is fuzzy normal if and only if it is constant on the conjugate classes of G and showed that the level subgroups of a fuzzy normal subgroup are all normal. Moreover, Wu in [6] introduced the concept of a normal fuzzy subgroup of a fuzzy group and used this concept to formulate the quotient structure of a fuzzy group. In [7], Mishref defined the Received February 1st, 2021; accepted March 3rd, 2021; published April 1st, 2021. 2010 Mathematics Subject Classification. 03F55, 03E72. Key words and phrases. intuitionistic fuzzy normed subrings; intuitionistic fuzzy normed normal subrings; direct product of intuitionistic fuzzy normed normal subrings. ©2021 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 341 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-341 Int. J. Anal. Appl. 19 (3) (2021) 342 maximal normal fuzzy subgroup and gave some of its properties in analogy to the crisp case, also he defined the subnormal, normal and composition series of normal fuzzy subgroups and explained the interrelationship between them and the series in the crisp case. In [8], a new type of fuzzy normal subgroups and fuzzy cosets was presented. The notion of intuitionistic fuzzy set was introduced by Atanassov [9] as a generalization of fuzzy sets. After that many researches applied this notion in various branches of mathematics especially in algebra and defined the notions of intuitionistic fuzzy subgroups, intuitionistic fuzzy subrings and intuitionistic fuzzy normal subgroups. Hur et al in [10], studied and characterized some properties of intuitionistic fuzzy normal subgroups of a group. In [11], Marashdeh and Salleh studied intuitionistic fuzzy groups by generalizing the notion of the fuzzy normal subgroup to the intuitionistic fuzzy normal subgroup. Later, some properties of intuitionistic fuzzy normal subrings were studied by Veeramani in [12], he described the algebraic nature of intuitionistic fuzzy normal subrings of a ring under homomorphism and anti-homomorphism. In [13], Sharma defined intuitionistic fuzzy magnified translation of intuitionistic fuzzy (normal) subring and ideal of a ring. The notion of intuitionistic fuzzy normed rings was introduced by Abed Alhaleem and Ahmad in [14]. The concepts of intuitionistic fuzzy normed subrings and intuitionistic fuzzy normed ideals was presented as an extension of fuzzy normed rings which were studied by Emniyet and Sahin in [15], which presented the notions of fuzzy normed subrings and fuzzy normed ideals. A generalization of normed ring was studied by Aren in [16]. Later, Naimark introduced normed rings in [17], which gave the first comprehensive treatment of Banach algebras. Gel’fand defined commutative normed rings in [18] and addressed them as complex Banach spaces with introduction of the notion of commutative normed rings. In this paper, we introduce the notion of intuitionistic fuzzy normed normal subrings. We study the algebraic nature of intuitionistic fuzzy normed normal subrings and characterize relevant properties. We introduce and study the notions of direct product of intuitionistic fuzzy normal subrings over normed rings. Further we define the relations between the intuitionistic characteristic function and intuitionistic fuzzy normed normal subrings. 2. Preliminaries In this section, we outline the most significant definitions and results needed for the following sections. We review some basic ideas of intuitionistic fuzzy set, intuitionistic fuzzy normed subring, definitions of normed linear spaces, t-norm and s-norm. Int. J. Anal. Appl. 19 (3) (2021) 343 Definition 2.1. [19] The fuzzy set A in a universal X is a set of ordered pairs: A = {(v,µA(v)) : v ∈ X} Where, µA(v) is the membership function of v in A which associates each element in X with a real number in the interval [0, 1]. Definition 2.2. [20] An intuitionistic fuzzy set (briefly, IFS) A in a nonempty set X is an object having the form IFS A = {(v,µA(v),γA(v) : v ∈ X}, where the functions µA(v) : X → [0, 1] and γA(v) : X → [0, 1] denote the degree of membership and the degree of nonmembership, respectively, where 0 ≤ µA(v)+γA(v)) ≤ 1 for all v ∈ X. An intuitionistic fuzzy set A is written symbolically in the form A = (µA,γA). Definition 2.3. [17] A ring A is said to be a normed ring (NR) if A possesses a norm ‖‖, that is, a non-negative real-valued function ‖‖ : A → R such that for any v,r ∈ A, (1) ‖v‖ = 0 ⇔ v = 0, (2) ‖v + r‖≤‖v‖ + ‖r‖ , (3) ‖v‖ = ‖−v‖, and (4) ‖vr‖≤‖v‖‖r‖. Definition 2.4. [14] Let A = {(v,µA(v),γA(v)) : v ∈ NR} and B = {(v,µB(v),γB(v)) : v ∈ NR} be two intuitionistic fuzzy normed rings over the normed ring NR. Then A is an intuitionistic fuzzy normed subring of B if µA(v) ≤ µB(v) and γA(v) ≥ γB(v). Proposition 2.5. [14] Let A be an intuitionistic fuzzy normed ring and let 0 be the zero of the normed ring NR, then the following is true for every v ∈ NR: (i) µA(v) ≤ µA(0) , γA(0) ≤ γA(v), (ii) µA(v) = µA(−v) , γA(v) = γA(−v). Lemma 2.6. [14] Let 1NR be the multiplicative identity of NR then for all v ∈ NR: 1. µA(v) ≥ µA(1NR) 2. γA(v) ≤ γA(1NR). Proposition 2.7. [21] Let A be an intuitionistic fuzzy set in a ring R, we denote the (α,β)-cut set by Aα,β = {v ∈ R : µA ≥ α and γA ≤ β} where α + β ≤ 1 and α,β ∈ [0, 1]. Definition 2.8. [22] Let ∗ : [0, 1] × [0, 1] → [0, 1] be a binary operation. Then ∗ is a t-norm if ∗ satisfies the conditions of commutativity, associativity, monotonicity and neutral element 1. We shortly use t-norm and write v∗r instead of ∗(v,r). Some examples of t-norm are v∗r= min {v,r} and v ∗r = v ·r. Int. J. Anal. Appl. 19 (3) (2021) 344 Definition 2.9. [23] Let � : [0, 1] × [0, 1] → [0, 1] be a binary operation. Then � is a s-norm if � satisfies the conditions of commutativity, associativity, monotonicity and neutral element 0. We shortly use s-norm and write v � r instead of �(v,r). Some examples of s-norm are v � r= max {v,r} and v �r = v + r −v ×r. 3. Some Properties of Intuitionistic Fuzzy Normed Normal Subrings Throughout the rest of this paper, R is the set of real numbers and NR is a normed ring. We define the intuitionistic fuzzy normed normal subrings and some basic properties related to it. Definition 3.1. [14] Let ∗ be a continuous t-norm and � be a continuous s-norm. An intuitionistic fuzzy set A = {(v,µA(v),γA(v)) : v ∈ NR} is called an intuitionistic fuzzy normed subring (IFNSR) of the normed ring (NR, +, .) if it satisfies the following conditions for all v,r ∈ NR: (i) µA(v −r) ≥ µA(v) ∗µA(r), (ii) µA(vr) ≥ µA(v) ∗µA(r), (iii) γA(v −r) ≤ γA(v) �γA(r), (iv) γA(vr) ≤ γA(v) �γA(r). Definition 3.2. Let NR be a normed ring. An intuitionistic fuzzy subring A of NR is said to be an intuitionistic fuzzy normed normal subring (IFNNSR) of NR if it satisfies the following for all v,r ∈ R: (i) µA(vr) = µA(rv), (ii) γA(vr) = γA(rv) Proposition 3.3. Let (NR, +, .) be a ring. If A and B are two intuitionistic fuzzy normed normal subrings of NR, then their intersection (A∩B) is an intuitionistic fuzzy normed normal subring of NR. Proof. Let v,r ∈ NR. Let A = {(v,µA(v),γA(v)) : v ∈ NR} and B = {(v,µB(v),γB(v)) : v ∈ NR} be intuitionistic fuzzy normed normal subrings. Let C = A ∩ B such that C = {(v,µC(v),γC(v)) : v ∈ NR} where µC(v)=min {µA(v),µB(v)} and γC(v)= max{γA(v),γB(v)}. µC(v −r) = min{µA(v −r),µB(v −r)} = µA(v −r) ∗µB(v −r) ≥{µA(v) ∗µA(r)}∗{µB(v) ∗µB(r)} = µA(v) ∗{µA(r) ∗µB(v)}∗µB(r) = µA(v) ∗{µB(v) ∗µA(r)}∗µB(r) = {µA(v) ∗µB(v)}∗{µA(r) ∗µB(r)} = µC(v) ∗µC(r) Int. J. Anal. Appl. 19 (3) (2021) 345 and µC(vr) = min{µA(vr),µB(vr)} = µA(vr) ∗µB(vr) ≥{µA(v) ∗µA(r)}∗{µB(v) ∗µB(r)} = µA(v) ∗{µA(r) ∗µB(v)}∗µB(r) = µA(v) ∗{µB(v) ∗µA(r)}∗µB(r) = {µA(v) ∗µB(v)}∗{µA(r) ∗µB(r)} = µC(v) ∗µC(r). Similarly γC(v −r) ≤ γC(v) �γC(r) and γC(vr) ≤ γC(v) �γC(r). Thus C is an intuitionistic fuzzy normed subring of NR. Now, µC(vr) = µA(vr) ∗µB(vr) = µA(rv) ∗µB(rv) = µC(rv). Therefore µC(vr) = µC(rv). Also γC(vr) = γA(vr) �γB(vr) = γA(rv) �γB(rv) = γC(rv). Therefore γC(vr) = γC(rv). Then, the intersection of any two intuitionistic fuzzy normed normal subrings is an intuitionistic fuzzy normed normal subring of NR. � Definition 3.4. Let A be a non-empty subset of the normed ring NR, the intuitionistic characteristic func- tion of A is defined as λA = (µλA,γλA ), where µλA (r) =   1 , if r ∈ A0 , if r /∈ A ,γλA (r) =   0 , if r ∈ A1 , if r /∈ A Lemma 3.5. If A = (µA,γA) is a subring of NR then λA = (µλA,γλA ) is an intuitionistic fuzzy normed normal subring of NR. Int. J. Anal. Appl. 19 (3) (2021) 346 Proof. It shown in [14] that λA = (µλA,γλA ) is an intuitionistic fuzzy normed subring of NR when A is a subring. Now, we need to show that λA = (µλA,γλA ) is an intuitionistic fuzzy normed normal subring, since vr and rv are in A, it follows that µλA (vr) = 1 = µλA (rv) and γλA (vr) = 0 = γλA (rv). Consequently, µλA (vr) = µλA (rv) and γλA (vr) = γλA (rv). Thus the intuitionistic characteristic function λA = (µλA,γλA ) of A is an intuitionistic fuzzy normed normal subring of NR. � Lemma 3.6. If A and B are two subrings of the ring NR, then their intersection A∩B is a subring of NR if and only if the intuitionistic characteristic function λC = (µλC,γλC ) of C = A ∩ B is an intuitionistic fuzzy normed normal subring of NR. Proof. Let C = A ∩ B be a subring of NR and v,r ∈ NR. If v,r ∈ C, then by definition of intuitionistic characteristic function µλC (v) = 1 = µλC (r) and γλC (v) = 0 = γλC (r). Since v−r, vr in A and B, it follows that v−r , vr in C. Thus, µλC (r−v) = 1 = 1∗1 = µλC (r)∗µλC (v) and µλC (rv) = 1 = 1∗1 = µλC (r)∗µλC (v). Thus µλC (r − v) ≥ µλC (r) ∗ µλC (v) and µλC (rv) ≥ µλC (r) ∗ µλC (v). Now γλC (v − r) = 0 = 0 � 0 = γλC (v) � γλC (r) and γλC (vr) = 0 = 0 � 0 = γλC (v) � γλC (r). Thus, γλC (v − r) ≤ γλC (v) � γλC (r) and γλC (vr) ≤ γλC (v) � γλC (r). As vr and rv ∈ C, so µλC (rv) = 1 = µλC (vr) and γλC (vr) = 0 = γλC (rv). Accordingly, µλC (rv) = µλC (vr) and γλC (vr) = γλC (rv). Similarly we have when v,r /∈ C: µλC (v −r) ≥ µλC (v) ∗µλC (r) and µλC (vr) ≥ µλC (v) ∗µλC (r), γλC (v −r) ≤ γλC (v) �γλC (r) and γλC (vr) ≤ γλC (v) �γλC (r), µλC (vr) = µλC (rv) and γλC (vr) = γλC (rv). Hence the intuitionistic characteristic function λC = (µλC,γλC ) of C is an intuitionistic fuzzy normed normal subring of NR. Conversely, assume that the intuitionistic characteristic function λC = (µλC,γλC ) of C is an intuitionistic fuzzy normal normed subring of NR. Let v,r ∈ C, this imply that µλC (v) = 1 = µλC (r) and γλC (v) = 0 = γλC (r), then: µλC (v −r) ≥ µλC (v) ∗µλC (r) = 1 ∗ 1 = 1, µλC (vr) ≥ µλC (v) ∗µλC (r) = 1 ∗ 1 = 1, γλC (v −r) ≤ γλC (v) �γλC (r) = 0 � 0 = 0, γλC (vr) ≤ γλC (v) �γλC (r) = 0 � 0 = 0. This implies that µλC (v − r) = 1, µλC (vr) = 1 and γλC (v − r) = 0, γλC (vr) = 0. Thus, v − r and vr ∈ C. Hence C is a subring of NR. � Proposition 3.7. If A is an intuitionistic fuzzy normed normal subring of a ring NR. Then 4A = (µA,µcA) is an intuitionistic fuzzy normed normal subring of a ring NR. Int. J. Anal. Appl. 19 (3) (2021) 347 Proof. Let v,r ∈ NR µcA(v −r) = 1 −µA(v −r) ≤ 1 − (µA(v) ∗µA(r)) ≤ 1 −min{µA(v),µA(r)} = max{1 −µA(v), 1 −µA(r)} = max{µcA(v),µ c A(r)}. Then, µcA(v −r) ≤ µ c A(v) �µ c A(r). µcA(vr) = 1 −µA(vr) ≤ 1 − (µA(v) ∗µA(r)) ≤ 1 −min{µA(v),µA(r)} = max{1 −µA(v), 1 −µA(r)} = max{µcA(v),µ c A(r)}. Then, µcA(vr) ≤ µ c A(v) �µ c A(r). Also, µcA(vr) = 1 −µA(vr) = 1 −µA(rv) = µ c A(rv), then µ c A(vr) = µ c A(rv). Therefore, 4A = (µA,µcA) is an intuitionistic fuzzy normed normal subring of NR. � Proposition 3.8. If A is an intuitionistic fuzzy normed normal subring of a ring NR. Then ♦A = (γcA,γA) is an intuitionistic fuzzy normed normal subring of a ring NR. Proof. Let v,r ∈ NR γcA(v −r) = 1 −γA(v −r) ≥ 1 − (γA(v) �γA(r)) ≥ 1 −max{γA(v),γA(r)} = min{1 −γA(v), 1 −γA(r)} = min{γcA(v),γ c A(r)}. Then, γcA(v −r) ≥ γ c A(v) ∗γ c A(r). γcA(vr) = 1 −γA(vr) ≥ 1 − (γA(v) �γA(r)) ≥ 1 −max{γA(v),γA(r)} = min{1 −µA(v), 1 −γA(r)} = min{γcA(v),γ c A(r)}. Then, γcA(vr) ≥ γ c A(v) ∗γ c A(r). Also, γcA(vr) = 1 −γA(vr) = 1 −γA(rv) = γ c A(rv), then γ c A(vr) = γ c A(rv). Therefore, ♦A = (γcA,γA) is an intuitionistic fuzzy normed normal ideal of NR. � Int. J. Anal. Appl. 19 (3) (2021) 348 Proposition 3.9. If A is an intuitionistic fuzzy normed normal subring of a ring NR. Then A = (µA,γA) is an intuitionistic fuzzy normed normal subring of NR if the fuzzy subsets µA and γ c A are intuitionistic fuzzy normed normal subrings of NR. Proof. Clearly, µA is an intuitionistic fuzzy normed normal subring of NR, we need to show that γA is an intuitionistic fuzzy normed normal subring of NR. 1 −γA(v −r) = γcA(v −r) ≥ γcA(v) ∗γ c A(r) ≥ min{γcA(v),γ c A(r)} = min{1 −γA(v), 1 −γA(r)} = 1 −max{γA(v),γA(r)}. Then, γA(v −r) ≤ γA(v) �γA(r). 1 −γA(vr) = γcA(vr) ≥ γcA(v) ∗γ c A(r) ≥ min{γcA(v),γ c A(r)} = min{1 −γA(v), 1 −γA(r)} = 1 −max{γA(v),γA(r)}. Then, γA(vr) ≤ γA(v) �γA(r). Also, 1 −γA(vr) = γcA(vr) = γ c A(rv) = 1 −γA(rv). Then, γA(vr) = γA(rv). Hence, A = (µA,γA) is an intuitionistic fuzzy normed normal subring of NR. � Proposition 3.10. If A is an intuitionistic fuzzy normed normal subring of a ring NR. Then A = (µA,γA) is an intuitionistic fuzzy normed normal subring of NR if the fuzzy subsets µcA and γA are intuitionistic fuzzy normed normal subrings of NR. Proof. Clearly, γA is an intuitionistic fuzzy normed normal subring of NR. We need to show that µA is an intuitionistic fuzzy normed normal subring of NR. 1 −µA(v −r) = µcA(v −r) ≤ µcA(v) �µ c A(r) ≤ max{µcA(v),µ c A(r)} = max{1 −µA(v), 1 −µA(r)} = 1 −min{µA(v),γA(r)}. Int. J. Anal. Appl. 19 (3) (2021) 349 Then, µA(v −r) ≥ µA(v) ∗µA(r). 1 −µA(vr) = µcA(vr) ≤ µcA(v) �µ c A(r) ≤ max{µcA(v),µ c A(r)} = max{1 −µA(v), 1 −µA(r)} = 1 −min{µA(v),µA(r)}. Then, µA(vr) ≥ µA(v) ∗µA(r). Also, 1 −µA(vr) = µcA(vr) = µ c A(rv) = 1 −µA(rv). Then, µA(vr) = µA(rv) Hence, A = (µA,γA) is an intuitionistic fuzzy normed normal subring of NR. � 4. Direct product of intuitionistic fuzzy normed normal subrings In this section, we define the direct product of intuitionistic fuzzy sets A1,A2 of normed rings R1,R2, respectively and examine some fundamental properties of direct product of intuitionistic fuzzy normed normal subrings. If NR1,NR2 are normed rings, then the direct product NR1 × NR2 of NR1 and NR2 is a normed ring with addition + defined as (v,r) + (z,d) = (v + z,r + d) and multiplication ◦ defined as (v,r) ◦ (z,d) = (vz,rd) for every (v,r), (z,d) in NR1 ×NR2. Definition 4.1. An intuitionistic fuzzy set (IFS) A×B = (µA×B,γA×B) of NR1×NR2 is an intuitionistic fuzzy normed subring (IFNSR) of NR1×NR2 if for all v = (v1,v2) and r = (r1,r2) in NR1×NR2, satisfies: (i) µA×B(v −r) ≥ µA×B(v) ∗µA×B(r); (ii) µA×B(vr) ≥ µA×B(v) ∗µA×B(r); (iii) γA×B(v −r) ≤ γA×B(v) �γA×B(r); (iv) γA×B(vr) ≤ γA×B(v) �γA×B(r). Definition 4.2. An intuitionistic fuzzy normed subring A×B = (µA×B,γA×B) of ring NR1 ×NR2 is an intuitionistic fuzzy normed normal subring of R1 ×R2 if for all v = (v1,v2) and r = (r1,r2) in R1 ×R2: µA×B(vr) = µA×B(rv) and γA×B(vr) = γA×B(rv). Lemma 4.3. If A and B are intuitionistic fuzzy normed subrings of the rings NR1 and NR2, respectively, then A × B is an intuitionistic fuzzy normed subring of the ring NR1 × NR2 under the same operations defined in NR1 ×NR2. Let A and B be two intuitionistic fuzzy normed subsets of NR1 and NR2, respectively. The direct product of A and B, is denoted by A×B, and defined as A×B = {((v,r),µA×B(v,r),γA×B(v,r)): for all v ∈ NR1 and r ∈ NR2} Int. J. Anal. Appl. 19 (3) (2021) 350 where µA×B(v,r) = min{µA(v),µB(r)} and γA×B(v,r) = max{γA(v),γB(r)}. Lemma 4.4. If A and B are intuitionistic fuzzy normed normal subrings of rings NR1 and NR2, respec- tively, then A×B is also an intuitionistic fuzzy normed normal subring NR1 ×NR2. Proof. Since the direct product of A and B is denoted by A × B = (µA×B,γA×B). Let (v,r), (z,d) be in NR1 ×NR2, then: µA×B((v,r) − (z,d)) = µA×B(v −z,r −d) = min{µA(v −z),µB(r −d)} = µA(v −z) ∗µB(r −d) ≥{µA(v) ∗µA(z)}∗{µB(r) ∗µB(d)} = µA(v) ∗{µA(z) ∗µB(r)}∗µB(d) = µA(v) ∗{µB(r) ∗µA(z)}∗µB(d) = {µA(v) ∗µB(r)}∗{µA(z) ∗µB(d)} = µA×B(v,r) ∗µA×B(z,d) and µA×B((v,r) ◦ (z,d)) = µA×B(vz,rd) = min{µA(vz),µB(rd)} = µA(vz) ∗µB(rd) ≥{µA(v) ∗µA(z)}∗{µB(r) ∗µB(d)} = µA(v) ∗{µA(z) ∗µB(r)}∗µB(d) = µA(v) ∗{µB(r) ∗µA(z)}∗µB(d) = {µA(v) ∗µB(r)}∗{µA(z) ∗µB(d)} = µA×B(v,r) ∗µA×B(z,d). Therefore, A×B is an intuitionistic fuzzy normed subring of NR1 ×NR2. Now, µA×B((v,r) ◦ (z,d)) = µA×B(vz,rd) = min{µA(vz),µB(rd)} = min{µA(zv),µB(dr)} = µA×B(zv,dr) = µA×B((z,d) ◦ (v,r)). Similarly, γA×B((v,r) − (z,d)) ≤ γA×B(v,r) �γA×B(z,d), γA×B((v,r) ◦ (z,d)) ≤ γA×B(v,r) �γA×B(z,d). and γA×B((v,r) ◦ (z,d)) = γA×B((z,d) ◦ (v,r)). Hence, A×B is an intuitionistic fuzzy normed normal subring of NR1 ×NR2. � Int. J. Anal. Appl. 19 (3) (2021) 351 Proposition 4.5. Let A and B be an intuitionistic fuzzy subsets of the rings NR1 and NR2 with identities 1NR1 and 1NR2 , respectively. If A×B is an intuitionistic fuzzy normed subring of NR1×NR2, then at least one of the following must holds: (i) µA(v) ≤ µB(1NR2 ) and γA(v) ≥ γB(1NR2 ); for all v ∈ NR1, (ii) µB(r) ≤ µA(1NR1 ) and γB(r) ≥ γA(1NR1 ); for all r ∈ NR2. Proof. Let A×B be an intuitionistic fuzzy normed subring of NR1 ×NR2, and let the statements (i) and (ii) does not holds, we can find v ∈ NR1 and r ∈ NR2 such that µA(v) > µB(1NR2 ), γA(v) < γB(1NR2 ) and µB(r) > µA(1NR1 ), γB(r) < γA(1NR1 ). Thus we have µA×B(vr) = min{µA(v),µB(r)} > min{µA(1NR1 ),µB(1NR2 )} = µA×B(1NR1, 1NR2 ) and γA×B(vr) = max{γA(v),γB(r)} < max{γA(1NR1 ),γB(1NR2 )} = γA×B(1NR1, 1NR2 ). which implies that A×B is not an intuitionistic fuzzy normed subring of NR1×NR2 which a contradiction. Therefore, at least one of the statements must hold. � Lemma 4.6. Let A and B be an intuitionistic fuzzy subsets of the rings NR1 and NR2 with identities 1NR1 and 1NR2 , respectively. If A×B is an intuitionistic fuzzy normed normal subring of NR1 ×NR2, then the following are true: (i) if µA(v) ≤ µB(1NR2 ) and γA(v) ≥ γB(1NR2 ), then A is an intuitionistic fuzzy normed normal subring of NR1. (ii) if µB(v) ≤ µA(1NR1 ) and γB(v) ≥ γA(1NR1 ), then B is an intuitionistic fuzzy normed normal subring of NR2. Proof. Let A × B be an intuitionistic fuzzy normed normal subring of NR1 × NR2 with v,r ∈ NR1 and 1NR2 ∈ NR2. Then (v, 1NR2 ) and (r, 1NR2 ) are in NR1 × NR2. Obviously, A is an intuitionistic fuzzy normed subring of NR1, then Int. J. Anal. Appl. 19 (3) (2021) 352 i. µA(v −r) = µA(v + (−r)) = min{µA(v + (−r)),µB(1NR2 + (−1NR2 ))} = µA×B((v + (−r)), (1NR2 + (−1NR2 )) = µA×B((v, 1NR2 ) + (−r,−1NR2 )) = µA×B((v, 1NR2 ) − (r, 1NR2 )) ≥ µA×B(v, 1NR2 ) ∗µA×B(r, 1NR2 ) = min{µA(v),µB(1NR2 )}∗min{µA(r),µB(1NR2 )} = µA(v) ∗µA(r). Also, µA(vr) = min{µA(vr),µB(1NR2 1NR2 )} = µA×B(vr, 1NR2 1NR2 ) = µA×B((v, 1NR2 ) ◦ (r, 1NR2 )) ≥ µA×B(v, 1NR2 ) ∗µA×B(r, 1NR2 ) = min{µA(v),µB(1NR2 )}∗min{µA(r),µB(1NR2 )} = µA(v) ∗µA(r) and with, µA(vr) = min{µA(vr),µB(1NR2 1NR2 )} = µA×B((vr), (1NR2 1NR2 )) = µA×B((v, 1NR2 ) ◦ (r, 1NR2 )) = µA×B((r, 1NR2 ) ◦ (v, 1NR2 )) = µA×B((rv), (1NR2 1NR2 )) = min{µA(rv),µB(1NR2 1NR2 )} = µA(rv). Similarly, we can prove that γA(v−r) ≤ γA(v) �γA(r), γA(vr) ≤ γA(v) �γA(r) and γA(vr) = γA(rv) for all v,r ∈ NR1. Hence, A is an intuitionistic fuzzy normed normal subring of NR1. ii. The proof is similar to the above. � Definition 4.7. Let A×B be a non-empty subset of the ring NR1 ×NR2. The intuitionistic characteristic function of A×B is denoted by λA×B = (µλA×B,γλA×B ) and defined as: µλA×B (r) =   1 , if r ∈ A×B0 , if r /∈ A×B ,γλA×B (r) =   0 , if r ∈ A×B1 , if r /∈ A×B Theorem 4.8. Let A and B be two subrings of the rings NR1 and NR2, respectively. Then A × B is a subring of NR1×NR2 if and only if the intuitionistic characteristic function λC = (µλC,γλC ) of C = A×B is an intuitionistic fuzzy normed normal subring of NR1 ×NR2. Int. J. Anal. Appl. 19 (3) (2021) 353 Proof. Let C = A × B be a subring of NR1 × NR2 and v,r ∈ NR1 × NR2. If v,r ∈ C = A × B, then by definition of intuitionistic characteristic function µλC (v) = 1 = µλC (r) and γλC (v) = 0 = γλC (r). Since v − r and vr ∈ C and C is a subring. It follows that µλC (v − r) = 1 = 1 ∗ 1 = µλC (v) ∗ µλC (r) and µλC (vr) = 1 = 1∗1 = µλC (v)∗µλC (r). Thus µλC (v−r) ≥ µλC (v)∗µλC (r) and µλC (vr) ≥ µλC (v)∗µλC (r). Now γλC (v − r) = 0 = 0 � 0 = γλC (v) � γλC (r) and γλC (v − r) = 0 = 0 � 0 = γλC (v) � γλC (r). Thus γλC (v −r) ≤ γλC (v) �γλC (r) and γλC (vr) ≤ γλC (v) �γλC (r). As vr and rv ∈ C, so µλC (vr) = 1 = µλC (rv) and γλC (vr) = 0 = γλC (rv). This implies that µλC (vr) = µλC (rv) and γλC (vr) = γλC (rv). Similarly we have µλC (v −r) ≥ µλC (v) ∗µλC (r) and µλC (vr) ≥ µλC (v) ∗µλC (r), γλC (v −r) ≤ γλC (v) �γλC (r) and γλC (vr) ≤ γλC (v) �γλC (r), µλC (vr) = µλC (rv) and γλC (vr) = γλC (rv). when v,r /∈ C. Hence the intuitionistic characteristic function λC = (µλC,γλC ) of C = A × B is an intu- itionistic fuzzy normed normal subring of NR1 ×NR2. On the other hand, assume that the intuitionistic characteristic function λC = (µλC,γλC ) of C = A×B is an intuitionistic fuzzy normed normal subring of NR1 × NR2. Now we have to show that C = A × B is a subring of NR. Let v,r ∈ C, where v = (v ′ ,r ′ ) and r = (v ′′ ,r ′′ ), where v ′ ,v ′′ ∈ A and r ′ ,r ′′ ∈ B. By definition µλC (v) = 1 = µλC (r) and γλC (v) = 0 = γλC (r), µλC (v −r) ≥ µλC (v) ∗µλC (r) = 1 ∗ 1 = 1, µλC (vr) ≥ µλC (v) ∗µλC (r) = 1 ∗ 1 = 1, γλC (v −r) ≤ γλC (v) �γλC (r) = 0 � 0 = 0, γλC (vr) ≤ γλC (v) �γλC (r) = 0 � 0 = 0. This implies that µλC (v − r) = 1, µλC (vr) = 1 and γλC (v − r) = 0, γλC (vr) = 0. Thus v − r and vr ∈ C. Hence C = A×B is a subring of NR1 ×NR2. � Lemma 4.9. If V = A×B and Q = C ×D are two subrings of NR1 ×NR2 then their intersection V ∩Q is also a subring of NR1 ×NR2. Theorem 4.10. Let V = A×B and Q = C×D be two intuitionistic fuzzy normed subrings of NR1×NR2. Then V ∩Q is subring of NR1×NR2 if and only if the intuitionistic characteristic function λZ = (µλZ,γλZ ) of Z = V ∩Q is an intuitionistic fuzzy normed normal subring of NR1 ×NR2. Proof. Let Z = V ∩ Q be a subring of ring NR1 × NR2 and let v = (v1,v2), r = (r1,r2) ∈ NR1 × NR2. If v,r ∈ Z = V ∩ Q, then by properties of intuitionistic characteristic function µλZ (v) = 1 = µλZ (r) and γλZ (v) = 0 = γλZ (r). Since v−r and vr ∈ Z. Then, µλZ (v−r) = 1 = 1∗1 = µλZ (v)∗µλZ (r), µλZ (vr) = 1 = Int. J. Anal. Appl. 19 (3) (2021) 354 1∗1 = µλZ (v)∗µλZ (r) and γλZ (v−r) = 0 = 0�0 = γλZ (v)�γλZ (v), γλZ (vr) = 0 = 0�0 = γλZ (v)�γλZ (r). Therefore, µλZ (v −r) ≥ µλZ (v) ∗µλZ (r), µλZ (vr) ≥ µλZ (v) ∗µλZ (r), γλZ (v −r) ≤ γλZ (v) �γλZ (r), γλZ (vr) ≤ γλZ (v) �γλZ (r). Since, vr and rv ∈ Z, then µλZ (vr) = 1 = µλZ (rv) and γλZ (vr) = 0 = γλZ (rv) so µλZ (vr) = µλZ (rv) and γλZ (vr) = γλZ (rv). We also have when v,r /∈ Z: µλC (v −r) ≥ µλC (v) ∗µλC (r) and µλC (vr) ≥ µλC (v) ∗µλC (r), γλC (v −r) ≤ γλC (v) �γλC (r) and γλC (vr) ≤ γλC (v) �γλC (r), µλC (vr) = µλC (rv) and γλC (vr) = γλC (rv). Hence the intuitionistic characteristic function λZ = (µλZ,γλZ ) of Z is an intuitionistic fuzzy normed normal subring of the ring NR1 ×NR2. Conversely, assume that the intuitionistic characteristic function λZ = (µλZ,γλZ ) is an intuitionistic fuzzy normed normal subring. Let v,r ∈ Z = V ∩Q, then µλZ (v) = 1 = µλZ (r) and γλZ (v) = 0 = γλZ (r), hence: µλC (v −r) ≥ µλC (v) ∗µλC (r) = 1 ∗ 1 = 1, µλC (vr) ≥ µλC (v) ∗µλC (r) = 1 ∗ 1 = 1, γλC (v −r) ≤ γλC (v) �γλC (r) = 0 � 0 = 0, γλC (vr) ≤ γλC (v) �γλC (r) = 0 � 0 = 0. Thus µλC (v − r) = 1 = µλC (vr) and γλC (v − r) = 0 = γλC (vr). This implies that v − r and vr ∈ Z. Hence Z is a subring of ring NR1 ×NR2. � Proposition 4.11. If the IFS A×B is an intuitionistic fuzzy normed normal subring of the ring NR1×NR2, then 4A×B = (µA×B,µcA×B) is an intuitionistic fuzzy normed normal subring of the ring NR1 ×NR2. Proof. Let A × B be an intuitionistic fuzzy normed normal subring of NR1 × NR2 and let (v,r), (z,d) ∈ NR1 ×NR2. Then µcA×B((v,r) − (z,d)) = 1 −µA×B((v,r) − (z,d)) ≤ 1 − (µA×B(v,r) ∗µA×B(z,d)) = 1 −min{µA×B(v,r),µA×B(z,d)} = max{1 −µA×B(v,r), 1 −µA×B(z,d)} = max{µcA×B(v,r),µ c A×B(z,d)} = µcA×B(v,r) �µ c A×B(z,d) Int. J. Anal. Appl. 19 (3) (2021) 355 and µcA×B((v,r) ◦ (z,d)) = 1 −µA×B((v,r) ◦ (z,d)) ≤ 1 − (µA×B(v,r) ∗µA×B(z,d)) = 1 −min{µA×B(v,r),µA×B(z,d)} = max{1 −µA×B(v,r), 1 −µA×B(z,d)} = max{µcA×B(v,r),µ c A×B(z,d)} = µcA×B(v,r) �µ c A×B(z,d). Thus 4A×B = (µA×B,µcA×B) is an intuitionistic fuzzy normed subring NR1 ×NR2. µcA×B((v,r) ◦ (z,d)) = 1 −µA×B((v,r) ◦ (z,d)) = 1 −µA×B((z,d) ◦ (v,r)) = µcA×B((z,d) ◦ (v,r)) Hence, 4A×B = (µA×B,µcA×B) is an intuitionistic fuzzy normed normal subring of NR1 ×NR2. � Proposition 4.12. If the IFS A×B is an intuitionistic fuzzy normed normal subring of the ring NR1×NR2, then ♦A×B = (γcA×B,γA×B) is an intuitionistic fuzzy normed normal subring of the ring NR1 ×NR2. Proof. Similar to the proof of Proposition 4.11 � Corollary 4.13. An IFS A×B is an intuitionistic fuzzy normed normal subring of the ring NR1 ×NR2 if and only if 4A × B = (µA×B,µcA×B) (resp.♦A × B = (γ c A×B,γA×B)) is an intuitionistic fuzzy normed normal subring of the ring NR1 ×NR2. Theorem 4.14. An IFS A×B is an intuitionistic fuzzy normed normal subring of the ring NR1 ×NR2 if and only if the fuzzy subsets µA×B and γ c A×B are intuitionistic fuzzy normed normal subring of the ring NR1 ×NR2. Proof. Let A×B = (µA×B,γA×B) be an intuitionistic fuzzy normed normal subring of the ring NR1×NR2. This implies that µA×B is an intuitionistic fuzzy normed normal subring of NR1 ×NR2. We have to show that γcA×B is also an intuitionistic fuzzy normed normal subring of the ring NR1 ×NR2. Let (v,r), (z,d) ∈ NR1 ×NR2. Then γcA×B((v,r) − (z,d)) = 1 −γA×B((v,r) − (z,d)) ≥ 1 − (γA×B(v,r) �γA×B(z,d)) = 1 −max{γA×B(v,r),γA×B(z,d)} = min{1 −γA×B(v,r), 1 −γA×B(z,d)} = min{γcA×B(v,r),γ c A×B(z,d)} = γcA×B(v,r) ∗γ c A×B(z,d) Int. J. Anal. Appl. 19 (3) (2021) 356 and γcA×B((v,r) ◦ (z,d)) = 1 −γA×B((v,r) − (z,d)) ≥ 1 − (γA×B(v,r) �γA×B(z,d)) = 1 −max{γA×B(v,r),γA×B(z,d)} = min{1 −γA×B(v,r), 1 −γA×B(z,d)} = min{γcA×B(v,r),γ c A×B(z,d)} = γcA×B(v,r) ∗γ c A×B(z,d) Hence, γcA×B is also an intuitionistic fuzzy normed subring of the ring NR1 ×NR2. γcA×B((v,r) ◦ (z,d)) = 1 −γA×B((v,r) ◦ (z,d)) = 1 −γA×B((z,d) ◦ (v,r)) = γcA×B((z,d) ◦ (v,r)). Hence, γcA×B is an intuitionistic fuzzy normed normal subring of NR1 ×NR2. Conversely, suppose that µA×B and γ c A×B are intuitionistic fuzzy normed normal subring of the ring NR1 × NR2. We have to show that A × B = (µA×B,γA×B) is an intuitionistic fuzzy normed normal subring of the ring NR1 ×NR2. Then 1 −γA×B((v,r) − (z,d)) = γcA×B((v,r) − (z,d)) ≥ γcA×B(z,d) ∗γ c A×B(v,r) = min{γcA×B(z,d),γ c A×B(v,r)} = min{1 −γA×B(z,d), 1 −γA×B(v,r)} = 1 −max{γA×B(z,d),γA×B(v,r)} = 1 − (γA×B(z,d) �γA×B(v,r)) and 1 −γA×B((v,r) ◦ (z,d)) = γcA×B((v,r) ◦ (z,d)) ≥ γcA×B(z,d) ∗γ c A×B(v,r) = min{γcA×B(z,d),γ c A×B(v,r)} = min{1 −γA×B(z,d), 1 −γA×B(v,r)} = 1 −max{γA×B(z,d),γA×B(v,r)} = 1 − (γA×B(z,d) �γA×B(v,r)). Therefore, A×B = (µA×B,γA×B) is an intuitionistic fuzzy normed subring of the ring NR1 ×NR2. 1 −γA×B((v,r) ◦ (z,d)) = γcA×B((v,r) ◦ (z,d)) = γcA×B((z,d) ◦ (v,r)) = 1 −γA×B((z,d) ◦ (v,r)). Int. J. Anal. Appl. 19 (3) (2021) 357 Therefore, A × B = (µA×B,γA×B) is an intuitionistic fuzzy normed normal subring of the ring NR1 × NR2. � Theorem 4.15. An IFS A×B is an intuitionistic fuzzy normed normal subring of the ring NR1 ×NR2 if and only if the fuzzy subsets µcA×B and γA×B are intuitionistic fuzzy normed normal subring of the ring NR1 ×NR2. Proof. Let A×B = (µA×B,γA×B) be an intuitionistic fuzzy normed normal subring of the ring NR1×NR2. This implies that γA×B is an intuitionistic fuzzy normed normal subring of NR1 ×NR2. We have to show that µcA×B is also an an intuitionistic fuzzy normed normal subring of the ring NR1 ×NR2. The proof of the first part is similar to the first part of Proposition 4.11. Conversely, suppose that µc A×B and γA×B are intuitionistic fuzzy normed normal subring of the ring NR1 × NR2. We have to show that A×B = (µA×B,γA×B) is an intuitionistic fuzzy normed normal subring of the ring NR1 ×NR2. Then 1 −µA×B((v,r) − (z,d)) = µcA×B((v,r) − (z,d)) ≤ µcA×B(z,d) �µ c A×B(v,r) = max{µcA×B(z,d),µ c A×B(v,r)} = max{1 −µA×B(z,d), 1 −µA×B(v,r)} = 1 −min{µA×B(z,d),µA×B(v,r)} = 1 − (µA×B(z,d) ∗µA×B(v,r)) and 1 −µA×B((v,r) ◦ (z,d)) = µcA×B((v,r) ◦ (z,d)) ≤ µcA×B(z,d) �µ c A×B(v,r) = max{µcA×B(z,d),µ c A×B(v,r)} = max{1 −µA×B(z,d), 1 −µA×B(v,r)} = 1 −min{µA×B(z,d),µA×B(v,r)} = 1 − (µA×B(z,d) ∗µA×B(v,r)). Therefore, A×B = (µA×B,γA×B) is an intuitionistic fuzzy normed subring of the ring NR1 ×NR2. 1 −µA×B((v,r) ◦ (z,d)) = µcA×B((v,r) ◦ (z,d)) = µcA×B((z,d) ◦ (v,r)) = 1 −µA×B((z,d) ◦ (v,r)). Therefore, A × B = (µA×B,γA×B) is an intuitionistic fuzzy normed normal subring of the ring NR1 × NR2. � Int. J. Anal. Appl. 19 (3) (2021) 358 5. Conclusion The objective of this paper was to initiate the notion of intuitionistic fuzzy normed normal subrings and to establish some relevant properties. 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