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An Investigation of ℱ-Closure of Fuzzy Submodules of a Module

An Investigation of ℱ-Closure of Fuzzy Submodules of a Module FUZZY INFORMATION AND ENGINEERING 2019, VOL. 11, NO. 2, 212–220 https://doi.org/10.1080/16168658.2021.1886820 An Investigation of F-Closure of Fuzzy Submodules of a Module a b c Ujwal Medhi , Helen K. Saikia and Bijan Davvaz a b Department of Mathematics, Arya Vidyapeeth College, Guwahati, India; Department of Mathematics, Gauhati University, Guwahati, India; Department of Mathematics, Yazd University, Yazd, Iran ABSTRACT ARTICLE HISTORY Received 18 October 2017 In this paper we introduce the notion of F -closure of fuzzy submod- Revised 28 November 2017 ules of a module M. Our attempt is to investigate various characteris- Accepted 30 June 2018 tics of such F -closures. If F is a non- empty set of fuzzy left ideals of R and μ is a fuzzy submodule of M then the F-closure of μ is denoted KEYWORDS by Cl (μ).If F is weak closed under intersection then (1) F-closure Fuzzy submodule; fuzzy of μ exhibits the submodule character, (2) the intersection of F – clo- closure; fuzzy torsion sures of two fuzzy submodules equals the F -closure of intersection 2010 MATHEMATICS of the fuzzy submodules. If F is weak closed under intersection then SUBJECT the submodule property of F -closure implies that F is left closed. CLASSIFICATIONS Moreover, if F is inductive then F is a topological filter if and only if F 08A72; 16D10 is a fuzzy submodule of M. 1. Introduction Closure operators have played a vital role in a variety of areas of mathematics. In case of category of modules there are different closure operators like T-closed, T-honest, which have been studied by Fay and Joubert.These operators are useful in the study of various aspects of rings and modules. For example, Jara obtained the characterizations of rings of quotients using the honest operator. The theory of honest subgroups was developed by Abian and Rinehart in [1]. The concepts of isolated submodules, honest submodules are studied by Fay and Joebert, Jara in [2, 3]. For a skew field, the notions of isolated submodules and honest submodules coincide. The honest submodules lead to a new characterization of ore domain. Moreover following the theory developed by Fay and Joebert, the in terms of In the category of groups isolated subgroups are useful in the study of torsion-free groups. The concept of super honest submodules was introduced by Joubert and Schoeman [4]. Super honest submodules of quasi injective modules are studied by Cheng [5]. In this paper our attempt is to extend to the notions of honest and superhonest submodules to fuzzy submodules. We define the concepts like honest fuzzy submodules, fuzzy closure, fuzzy tor- sion and superhonest fuzzy submodules. Various characteristics of honest and superhonest submodules are fuzzified in this paper. CONTACT Helen K. Saikia hsaikia@yahoo.com © 2021 The Author(s). Published by Taylor & Francis Group on behalf of the Fuzzy Information and Engineering Branch of the Operations Research Society, Guangdong Province Operations Research Society of China. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons. org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. FUZZY INFORMATION AND ENGINEERING 213 2. Preliminaries Throughout this paper R is a non commutative ring with unity and M is a left R-module. The zero elements of R and M are 0 and θ, respectively. Definition 2.1: A fuzzy subset μ of R is called a fuzzy left ideal if it satisfies the following properties [10]: (1) μ(x − y) ≥ μ(x) ∧ μ(y), for all x, y ∈ R, (2) μ(xy) ≥ μ(y), for all x, y ∈ R. Definition 2.2: A fuzzy subset μ of R is called a fuzzy ideal if it satisfies the following properties [10]: (1) μ(x − y) ≥ μ(x) ∧ μ(y), for all x, y ∈ R, (2) μ(xy) ≥ μ(x) ∨ μ(y), for all x, y ∈ R. Definition 2.3: A fuzzy subset μ of M is called fuzzy submodules of M if the following conditions are satisfied: (1) μ(x − y) ≥ μ(x) ∧ μ(y), for all x, y ∈ M, (2) μ(rx) ≥ μ(x), for all r ∈ R, x ∈ M, (3) μ(θ ) = 1. The set of all fuzzy submodules is denoted by F(M). Definition 2.4 ([6]): Let μ be a fuzzy subset of a non-empty set X.For t ∈ [0, 1], μ ={x ∈ X | μ(x) ≥ t} is called t-cut set of μ. Definition 2.5: Let μ be a fuzzy subset of a non-empty set X. Then a fuzzy point x , x ∈ X, t ∈ (0, 1] is defined as the fuzzy subset x of X such that x (x) = t,and x (y) = 0, for all t t t y ∈ X − x. We use the notation x ∈ μ if and only if x ∈ μ . t t Definition 2.6 ([7]): Let Y be a subset of a non-empty set X and t ∈ (0, 1], then t is defined as t,if x ∈ Y t (x) = 0, otherwise When t = 1 then 1 , is known as the characteristic function of Y and it is denoted by χ . Y Y Definition 2.7 ([8]): A fuzzy left ideal μ of R is called a essential fuzzy left ideal of R, denoted by μ ⊆ R, if for every nonzero fuzzy left ideal δ of R, there exist x(= 0) ∈ R such that x ∈ μ e t and x ∈ δ , for some t ∈ (0, 1]. Definition 2.8 ([8]): Let μ and σ be two nonzero fuzzy left ideals of R such that μ ⊆ σ . Then μ is called a fuzzy essential left ideal in σ , denoted by μ ⊆ σ if for every nonzero e 214 U. MEDHI ET AL. fuzzy left ideal δ of R satisfying δ ⊆ σ , there exist x(= 0) ∈ R such that x ∈ μ and x ∈ δ,for t t some t ∈ (0, 1]. Lemma 2.9 ([8]): A fuzzy left ideal μ of R is a essential fuzzy left ideal of R if and only if μ is fuzzy essential left ideal of χ . Lemma 2.10 ([8]): Let μ, ν and σ be nonzero fuzzy left ideals of R such that μ ⊆ ν ⊆ σ . Then μ ⊆ σ if and only if μ ⊆ ν ⊆ σ . e e e Definition 2.11: Let μ be a fuzzy subset of a non empty set X. Then the support of μ, ∗ ∗ denoted by μ is defined as μ ={x ∈ X : μ(x)> 0}.If μ is a fuzzy submodule of M then μ is a submodule of M. Definition 2.12 ([7]): Let ζ be a fuzzy ideal and μ ∈ F(M). We define ζμ as follows: (ζ μ)(x) =∨{ζ(r) ∧ μ(y) | r ∈ R, y ∈ M, ry = x}, ∀ x ∈ M. Definition 2.13 ([9]): Let μ be a fuzzy subset of an R-module M. Then the fuzzy subset ann(μ) of R is defined as follows: ann(μ) = {η | η ∈ [0, 1] , ημ ⊆ χ }. Lemma 2.14 ([9]): Let μ ∈ [0, 1] . Then ann(μ) = {r | r ∈ R, α ∈ [0, 1], r μ ⊆ χ }. α α 0 Definition 2.15 ([7]): Let μ be a fuzzy submodule of M and ν be any fuzzy subset of M. Then (μ : ν) = {η | η ∈ [0, 1] , ην ⊆ μ}. Lemma 2.16 ([7]): Let μ, ν be any two fuzzy subsets of M. Then μ : ν = {r | r ∈ R, α ∈ [0, 1], r ν ⊆ ν}. α α Definition 2.17: Let f be a R-module homomorphism from M to M , μ ∈ F(M) and ν ∈ −1 F(M ). Then f (μ) ∈ F(M ) and f (ν) ∈ F(M) are defined by 1 1 −1 sup −1 μ(m),if f (w) = φ m∈f (w) f (μ)(w) = 0, otherwise −1 and f (ν)(m) = ν(f (m)), for all w ∈ M , m ∈ M. 3. Fuzzy Submodules In this section let F be a non empty set of fuzzy left ideals of R. FUZZY INFORMATION AND ENGINEERING 215 Definition 3.1: Let M be a left R-module and μ be a fuzzy submodule of M. Then we define fuzzy torsion of μ as follows: T(μ) = {γ | γ ∈ [0, 1] , γ ⊆ μ, γσ ⊆ χ , for some fuzzy ideal σ }. Also we define the F-torsion of μ as follows: M M T (μ) = {γ | γ ∈ [0, 1] , γ ⊆ μ there is σ ∈ F such that γσ ⊆ χ } M M 1 is F-torsion if T (1 ) = 1 and F-torsion free if T (1 ) = χ . M M M M θ F F Definition 3.2: Let M be a left R-module and μ be a fuzzy submodule of M. Then we define fuzzy closure of μ as follows: Cl(μ) = {σ | σ ∈ [0, 1] , σ ⊆ μ, γσ ⊆ μ, for some fuzzy ideal γ }. Also we define the F-closure of μ as follows: M M Cl (μ) = {σ | σ ∈ [0, 1] , σ ⊆ μ, there is γ ∈ F such that γσ ⊆ μ}. Lemma 3.3: T(μ) = {m | m ∈ μ, m σ ⊆ χ , for some fuzzy ideal σ } α α α θ Proof: Clearly, {m | m ∈ μ, m σ ⊆ χ , for some fuzzy ideal σ } α α α θ ⊆{γ | γ ∈ [0, 1] , γ ⊆ μ, γσ ⊆ χ , for some fuzzy ideal σ }. Therefore, {m ∈ μ, m σ ⊆ χ , for some fuzzy ideal σ } α α θ ⊆ {γ | γ ∈ [0, 1] , γ ⊆ μ, γσ ⊆ χ , for some fuzzy ideal σ }. = T(μ) Let γ ∈ [0, 1] such that γσ ⊆ χ , for some fuzzy ideal σ.Let m ∈ m such that γ(m) = α. Now,(m σ)(x) =∨{m (s) ∧ σ(y) | x = sy; s ∈ M, y ∈ R} α α =∨{γ(m) ∧ σ(y) | x = my; s ∈ M, y ∈ R} ≤{γ(s) ∧ σ(y) | x = sy; s ∈ M, y ∈ R} = (γ σ )(x) ⊆ χ (x). Thus (m σ) ⊆ χ . α θ Therefore, T(μ) ⊆ {m | m ∈ μ, m σ ⊆ χ , for some fuzzy ideal σ }. α α α θ Hence the result follows. The proofs of the following lemmas are similar. Lemma 3.4: T (μ) = {m | m ∈ μ there is γ ∈ F such that m γ ⊆ χ } α α α θ Lemma 3.5: Cl(μ) = {m | m ∈ μ, γ m ⊆ μ, for some fuzzy ideal γ } α α α Lemma 3.6: Cl (μ) = {m | m ∈ μ, there is γ ∈ F such that γ m ⊆ μ} α α α Definition 3.7: Let M be a left R-module and μ ∈ F(M).Wecall μ is F-closed if Cl (μ) = μ Definition 3.8: F is called weak closed under intersection if for any μ , μ ∈ F there exists 1 2 σ ∈ F such that σ ⊆ μ ∩ μ . 1 2 216 U. MEDHI ET AL. Definition 3.9: F is called inductive if for any μ ∈ F and any left ideal σ ⊇ μ,wehave σ ∈ F. Definition 3.10: F is called left closed if for any r ∈ 1 and any μ ∈ F, there is σ ∈ F such t R that σ r ⊆ μ i.e. (μ : r ) ⊆ σ . t t Definition 3.11: F is called a topological filter if it is closed under intersection, inductive and left closed. Theorem 3.12: If F is the set of all fuzzy essential ideals of R, then F is inductive. Proof: Let μ ∈ F.If σ is any fuzzy left ideal of R such that σ ⊇ μ. Then μ ⊆ R and so μ ⊆ e e 1 .Thusfrom μ ⊆ σ ⊆ 1 , it follows that σ ⊆ R and hence σ ∈ F. R R e Theorem 3.13: Let M be a left R-module. (a) If F is weak closed under intersection, then for any μ ∈ F(M) we have that Cl (μ) is a fuzzy submodule of M. M M M (b) If F is weak closed under intersection if and only if Cl (σ ) ∩ Cl (σ ) = Cl (σ ∩ σ ),for 1 2 1 2 F F F any two σ , σ ∈ F(M). 1 2 (c) If F is weak closed under intersection, then F is left closed if and only if Cl (σ ) is a fuzzy submodule of M for any σ ∈ F(M). Proof: (a) Let m , m ∈ M. 1 2 M M Now, Cl (μ)(m ) ∧ Cl (μ)(m ) 1 2 F F = ( {γ (m ) | γ ∈ [0, 1] , γ ⊆ μ, there is μ ∈ F such that μ γ ⊆ μ}) 1 1 1 1 1 1 1 ( {γ (m ) | γ ∈ [0, 1] , γ ⊆ μ, there is μ ∈ F such that μ γ ⊆ μ}) 2 2 2 1 2 2 2 = {γ (m ) ∧ γ (m ) | γ ∈ [0, 1] , γ ⊆ μ, there is 1 1 2 2 i i μ ∈ F such that μ γ ⊆ μ; i = 1, 2} i i i ≤{(γ + γ )(m ) ∧ (γ + γ )(m ) | γ ∈ [0, 1] , γ ⊆ μ, there is 1 2 1 1 2 2 i i μ ∈ F such that μ γ ⊆ μ; i = 1, 2} i i i ≤{(γ + γ )(m − m ) | μ(γ + γ )} 1 2 1 2 1 2 Since F is weak closed, it follows that μ , μ ∈ F implies there is σ ∈ F such that σ ⊆ 1 2 μ ∩ μ . Therefore we have, 1 2 σ(γ + γ ) ⊆ (μ ∩ μ )(γ + γ ) 1 2 1 2 1 2 ⊆ (μ ∩ μ )γ + (μ ∩ μ )γ 1 2 1 1 2 2 ⊆ μ γ + μ γ 1 1 2 2 ⊆ μ + μ = μ M M Thus, Cl (μ)(m ) ∧ Cl (μ)(m ) 1 2 F F ≤{(γ + γ )(m − m ) | σ(γ + γ )(m − m ) ⊆ μ, for some σ ∈ F} 1 2 1 2 1 2 1 2 = Cl (μ)(m − m ) 1 2 Also Cl (μ)(rm) = {γ(rm) | γ ∈ [0, 1] , γ ⊆ μ, there is σ ∈ F such that σγ ⊆ μ} FUZZY INFORMATION AND ENGINEERING 217 ≥ {γ(m) | γ ∈ [0, 1] , γ ⊆ μ, there is σ ∈ F such that σγ ⊆ μ} = Cl (μ)(m) Therefore Cl (μ) ∈ F(M). (b) (⇒). For any m ∈ M, Cl (σ ∩ σ )(m) 1 2 = {γ(m) | γ ∈ [0, 1] , γ ⊆ (σ ∩ σ ), there is σ ∈ F such that σγ ⊆ σ ∩ σ } 1 2 1 2 = {γ(m) ∧ γ(m) | γ ∈ [0, 1] , γ ⊆ (σ ∩ σ ), there is σ ∈ F such that 1 2 σγ ⊆ σ , σγ ⊆ σ } 1 2 = ( {γ(m) | γ ∈ [0, 1] , γ ⊆ σ , there is σ ∈ F such that σγ ⊆ σ }) 1 1 ( {γ(m) | γ ∈ [0, 1] , γ ⊆ σ , there is σ ∈ F such that σγ ⊆ σ }) 2 2 M M ≤ Cl (σ )(m) Cl (σ )(m) 1 2 F F M M = [Cl (σ ) ∩ Cl (σ )](m). 1 2 F F M M M Thus Cl (σ ∩ σ ) ⊆ Cl (σ ) ∩ Cl (σ ). 1 2 1 2 F F F M M Also, [Cl (σ ) ∩ Cl (σ )](m) 1 2 F F M M = Cl (σ )(m) Cl (σ )(m) 1 2 F F = ( {γ (m) | γ ∈ [0, 1] , γ ⊆ σ , there is μ ∈ F such that μ γ ⊆ σ }) 1 1 1 1 1 1 1 1 ( {γ (m) | γ ∈ [0, 1] , γ ⊆ σ , there is μ ∈ F such that μ γ ⊆ σ }) 2 2 2 2 2 2 2 2 = {γ (m) ∧ γ (m) | γ ∈ [0, 1] , γ ⊆ σ , there is μ ∈ F such that 1 2 i i i i μ γ ⊆ σ ; i = 1, 2} i i i = {(γ ∩ γ )(m) | γ ∈ [0, 1] , γ ⊆ σ , there is μ ∈ F such that 1 2 i i i i μ γ ∩ μ γ ⊆ σ ∩ σ ; i = 1, 2} Now, we have 1 1 2 2 1 2 (μ ∩ μ )(γ ∩ γ ) ⊆ (μ ∩ μ )γ ∩ (μ ∩ μ )γ 1 2 1 2 1 2 1 1 2 2 ⊆ γ μ ∩ γ μ 1 1 2 2 ⊆ σ ∩ σ 1 2 Since F is weak closed, for μ , μ ∈ F there exists μ ∈ F such that μ ⊆ μ ∩ μ . 1 2 1 2 Therefore, μ(γ ∩ γ ) ⊆ (μ ∩ μ )(γ ∩ γ ) ⊆ σ ∩ σ . 1 2 1 2 1 2 1 2 M M Thus we have, [Cl (σ ) ∩ Cl (σ )](m) 1 2 F F ≤ {γ(m) | γ ∈ [0, 1] , γ ⊆ (σ ∩ σ ), there is μ ∈ F 1 2 such that μγ ⊆ σ ∩ σ } 1 2 = Cl (σ ∩ σ ). Hence the result follows. 1 2 (⇐).Let μ , μ ∈ F. 1 2 We have, (μ 1 )(x) =∨{μ (y) ∧ 1 (z) | yz = x, y ∈ R}≤ μ (x) ∧ α ≤ μ (x). i α i α i i R R R Thus μ 1 ⊆ μ , ∀i = 1, 2 and therefore 1 ∈ [Cl (σ ) ∩ Cl (σ )] = Cl (σ ∩ σ ), hence i α i α 1 2 1 2 F F F there exists μ ∈ F such that μ ⊆ σ ∩ σ . 1 2 (c) (⇒). Follows from part (a). R R (⇐).Let μ ∈ F and r ∈ 1 , then Cl (μ) = 1 , hence r ∈ Cl (μ) and therefore there is t R R t F F ν ∈ F such that νr ⊆ μ. Theorem 3.14: Let F be an inductive set of fuzzy ideals, then the following statements are equivalent: (a) F is a topological filter. (b) Cl (σ ) is a fuzzy submodule for any σ ∈ F(M). F 218 U. MEDHI ET AL. Proof: (a) ⇒ (b). F closed under intersection, so weak closed under intersection. It is given that F is left closed, hence by part (c) of the above theorem the result follows. (b) ⇒ (a). F is weak closed under intersection and left closed as Cl (σ ) is a fuzzy sub- module for any σ ∈ F(M).Since F is inductive, therefore it is closed under intersection. Hence F is a topological filter. Definition 3.15: Let M be an R-module. Then a fuzzy submodule μ of M is said to F-closed in 1 if for any fuzzy ideal σ ∈ F and any m ∈ 1 , χ = σ m ⊆ μ implies m ∈ μ, α ∈ M α M θ α α (0, 1). Theorem 3.16: Let μ ⊆ σ ⊆ 1 .If μ is F-closed in σ , σ is F-closed in 1 then μ is F-closed in M M 1 . Proof: Let δ ∈ F and m ∈ 1 such that χ = m δ ⊆ μ.Thus m δ ⊆ μ ⊆ σ.Since σ is F- α M θ α α closed in 1 , it follows that m ∈ σ . Now, m ∈ σ and m δ ⊆ μ.Since μ is F-closed in σ,so M α α α it gives m ∈ μ. Hence μ is F-closedin1 . α M Theorem 3.17: Let σ ∈ F(M).If σ is F-closed in 1 and inductive then, for any m ∈ M α Cl (σ )\σ , we have (σ : m ) = Ann(m ). α α Proof: Suppose σ is F-closedin1 and inductive. Clearly Ann(m ) ⊆ (σ : m ). M α α Let m ∈ Cl (σ )\σ . Then there exists μ ∈ F such that μm ⊆ χ and this implies μ ⊆ α α θ Annm .Since F isinductive,sowehave Ann(m ) ∈ F.Now Ann(m ) ⊆ (σ : m ) implies α α α α (σ : m ) ∈ F. Next let r ∈ (σ : m ), then r m ⊆ σ , therefore (σ : m )m ⊆ σ and thus (σ : α t α t α α α m )m ⊆ χ . This implies (σ : m ) ⊆ Ann(m ). Hence (σ : m ) = Ann(m ). α α θ α α α α Theorem 3.18: Let σ ∈ F(M). For any m ∈ Cl (σ )\σ , we have (σ : m ) = Ann(m ), then α α α 1 m ∩ σ = χ , R α θ Proof: We have 1 m (x) =∨{1 (y) ∧ m (z) | y ∈ R, z ∈ M, x = yz} R α R α =∨{m (z) | y ∈ R, z ∈ M, x = yz} 0if x ∈ / Rm α if x ∈ Rm = (Rm) ∗ ∗ Then (Rm) is a fuzzy submodule. Let μ = (σ : m ) and L ={r ∈ R | rm ∈ σ }.Let x ∈ μ , α α then ∃α ∈ [0, 1] such that (xm) ∈ σ and therefore σ(xm) ≥ α ∧ p > 0. Consequently, α∧p ∗ ∗ xm ∈ σ and thus x ∈ L. Hence μ ⊆ L Again let x ∈ L, then x ∈ σ and this implies σ(xm)> 0. Let σ(xm) = α, then σ(xm)> α ∧ p,which gives (xm) ∈ σ.Thus x ∈ μ i.e. μ(x) ≥ α> 0 therefore x ∈ μ ,thus α∧p α ∗ ∗ ∗ ∗ ∗ ∗ L ⊆ μ . Hence μ = L.Thus (σ : m ) = L ={r | rm ∈ σ }= (σ : m).Also[Ann(m )] = α α ∗ ∗ Ann(m).Now,[(Rm) ∩ σ ] ={x | (Rm) ∧ σ(x)> 0}={x | x ∈ Rm and x ∈ σ }= Rm ∩ α α ∗ ∗ ∗ σ .By hypothesis we have, (σ : m ) = Ann(m ),thisimplies (σ : m) = [Ann(m )] = α α α FUZZY INFORMATION AND ENGINEERING 219 ∗ ∗ Ann(m) and therefore Rm ∩ σ = 0. So, [(Rm) ∩ σ ] = 0. Hence (Rm) ∩ σ = 1 m ∩ α α R α σ = χ . Theorem 3.19: Let σ ∈ F(M). If for any m ∈ Cl (σ )\σ , we have 1 m ∩ σ = χ , thenμ is α R α θ F-closed. Proof: Let μ ∈ F, m ∈ 1 such that χ = μm ⊆ σ.If m ∈ / σ then we have, by hypoth- α M θ α α esis 1 m ∩ σ = χ .Now μm ⊆ σ implies μm ∩ σ = μm ⊆ 1 m . Therefore μm = R α θ α α α R α α μm ∩ σ ⊆ 1 m ∩ σ = χ , a contradiction. Hence the result follows. α R α θ As the consequences of the Theorems 3.17, 3.18, 3.19 we obtain the following: Theorem 3.20: Let σ ∈ F(M) and F be inductive. Then the following statements are equiva- lent: (a) σ is F-closed in 1 . (b) For any m ∈ Cl (σ )\σ , we have (σ : m ) = Ann(m ). α α α (c) For any m ∈ Cl (σ )\σ , we have 1 ∩ m = χ . α R α θ M M Theorem 3.21: Let σ ∈ F(M) be an F-closed, then Cl (σ ) = σ ∪ T (1 ). F F M M M Proof: Clearly σ ∪ T (1 ) ⊆ Cl (σ ). Now let m ∈ Cl (σ )\σ , there exists μ ∈ F such that M α F F F μm = χ ,thus m ∈ T (1 ). α θ α M Acknowledgments The authors are really thankful to the referee(s) for reading the article and for making valuable helpful comments. Disclosure statement No potential conflict of interest was reported by the author(s). Funding The authors did not receive any funding for this research. Notes on contributors Ujwal Medhi is an Assistant Professor in the Department of Mathematics at Arya Vidyapeeth College, Guwahati, India. He completed Ph.D. degree under the supervision of Prof. H. K. Saikia from Gauhati University, India. His area of interest is fuzzy algebra. Helen K Saikia is a Professor in the Department of Mathematics at Gauhati University, Guwahati, India. Her areas of research interest are number theory, algebra, fuzzy algebra and graph theory. Sixteen research scholars have obtained Ph.D. degree under direct supervision of her. She has number of publications in reputed international journals. Bijan Davvaz is a full professor in the Department of Mathematics at Yazd University, Yazd, Iran. He works on algebra, algebraic hyperstructures, rough sets and fuzzy logic. He is a member of editorial 220 U. MEDHI ET AL. boards for 25 mathematical journals. He has authored around 500 research papers, especially on alge- bra, algebraic hyperstructures and their applications and fuzzy logic. He has also published five books on algebra. References [1] Abian A, Reinhart D. Honest subgroups of abelian groups. Rend Circ Math Palermo. 1963;12:353–356. [2] Fay TH, Joubert SV. Isolated submodules and skew fields. Appl Categorial Struct. 2000;8:317–326. [3] Pascual J. Honest submodules. Czochoslovak Math J. 2007;57(132):225–241. [4] Jouvert SV, Schoeman MJ. Superhonesty for modules and abelian groups. J Math. 1984;12(2):87– [5] Cheng MH. Notes on superhonest submodules. Chinese J Math. 1986;14(2):109–119. [6] Zadeh LA. Fuzzy sets. Inf Control. 1965;8:338–353. [7] Mordeson JN, Malik DS. Fuzzy commutative algebra. Singapore: World Scientific; 1998. [8] Medhi U, Rajkhowa KK, Barthakur LK, et al. On fuzzy essential ideals of rings. Adv Fuzzy Sets Syst. 2008;3(3):287–299. [9] Kalita MC, Saikia HK. On annihilators of fuzzy subsets of modules. Int J Algebra. 2009;3(10):483– [10] Liu W. Fuzzy invariant subgroups and fuzzy ideals. Fuzzy Sets Syst. 1982;8:133–139. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Fuzzy Information and Engineering Taylor & Francis

An Investigation of ℱ-Closure of Fuzzy Submodules of a Module

An Investigation of ℱ-Closure of Fuzzy Submodules of a Module

Abstract

In this paper we introduce the notion of -closure of fuzzy submodules of a module M. Our attempt is to investigate various characteristics of such -closures. If is a non- empty set of fuzzy left ideals of R and μ is a fuzzy submodule of M then the -closure of μ is denoted by . If is weak closed under intersection then (1) -closure of μ exhibits the submodule character, (2) the intersection of – closures of two fuzzy submodules equals the -closure of intersection of...
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FUZZY INFORMATION AND ENGINEERING 2019, VOL. 11, NO. 2, 212–220 https://doi.org/10.1080/16168658.2021.1886820 An Investigation of F-Closure of Fuzzy Submodules of a Module a b c Ujwal Medhi , Helen K. Saikia and Bijan Davvaz a b Department of Mathematics, Arya Vidyapeeth College, Guwahati, India; Department of Mathematics, Gauhati University, Guwahati, India; Department of Mathematics, Yazd University, Yazd, Iran ABSTRACT ARTICLE HISTORY Received 18 October 2017 In this paper we introduce the notion of F -closure of fuzzy submod- Revised 28 November 2017 ules of a module M. Our attempt is to investigate various characteris- Accepted 30 June 2018 tics of such F -closures. If F is a non- empty set of fuzzy left ideals of R and μ is a fuzzy submodule of M then the F-closure of μ is denoted KEYWORDS by Cl (μ).If F is weak closed under intersection then (1) F-closure Fuzzy submodule; fuzzy of μ exhibits the submodule character, (2) the intersection of F – clo- closure; fuzzy torsion sures of two fuzzy submodules equals the F -closure of intersection 2010 MATHEMATICS of the fuzzy submodules. If F is weak closed under intersection then SUBJECT the submodule property of F -closure implies that F is left closed. CLASSIFICATIONS Moreover, if F is inductive then F is a topological filter if and only if F 08A72; 16D10 is a fuzzy submodule of M. 1. Introduction Closure operators have played a vital role in a variety of areas of mathematics. In case of category of modules there are different closure operators like T-closed, T-honest, which have been studied by Fay and Joubert.These operators are useful in the study of various aspects of rings and modules. For example, Jara obtained the characterizations of rings of quotients using the honest operator. The theory of honest subgroups was developed by Abian and Rinehart in [1]. The concepts of isolated submodules, honest submodules are studied by Fay and Joebert, Jara in [2, 3]. For a skew field, the notions of isolated submodules and honest submodules coincide. The honest submodules lead to a new characterization of ore domain. Moreover following the theory developed by Fay and Joebert, the in terms of In the category of groups isolated subgroups are useful in the study of torsion-free groups. The concept of super honest submodules was introduced by Joubert and Schoeman [4]. Super honest submodules of quasi injective modules are studied by Cheng [5]. In this paper our attempt is to extend to the notions of honest and superhonest submodules to fuzzy submodules. We define the concepts like honest fuzzy submodules, fuzzy closure, fuzzy tor- sion and superhonest fuzzy submodules. Various characteristics of honest and superhonest submodules are fuzzified in this paper. CONTACT Helen K. Saikia hsaikia@yahoo.com © 2021 The Author(s). Published by Taylor & Francis Group on behalf of the Fuzzy Information and Engineering Branch of the Operations Research Society, Guangdong Province Operations Research Society of China. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons. org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. FUZZY INFORMATION AND ENGINEERING 213 2. Preliminaries Throughout this paper R is a non commutative ring with unity and M is a left R-module. The zero elements of R and M are 0 and θ, respectively. Definition 2.1: A fuzzy subset μ of R is called a fuzzy left ideal if it satisfies the following properties [10]: (1) μ(x − y) ≥ μ(x) ∧ μ(y), for all x, y ∈ R, (2) μ(xy) ≥ μ(y), for all x, y ∈ R. Definition 2.2: A fuzzy subset μ of R is called a fuzzy ideal if it satisfies the following properties [10]: (1) μ(x − y) ≥ μ(x) ∧ μ(y), for all x, y ∈ R, (2) μ(xy) ≥ μ(x) ∨ μ(y), for all x, y ∈ R. Definition 2.3: A fuzzy subset μ of M is called fuzzy submodules of M if the following conditions are satisfied: (1) μ(x − y) ≥ μ(x) ∧ μ(y), for all x, y ∈ M, (2) μ(rx) ≥ μ(x), for all r ∈ R, x ∈ M, (3) μ(θ ) = 1. The set of all fuzzy submodules is denoted by F(M). Definition 2.4 ([6]): Let μ be a fuzzy subset of a non-empty set X.For t ∈ [0, 1], μ ={x ∈ X | μ(x) ≥ t} is called t-cut set of μ. Definition 2.5: Let μ be a fuzzy subset of a non-empty set X. Then a fuzzy point x , x ∈ X, t ∈ (0, 1] is defined as the fuzzy subset x of X such that x (x) = t,and x (y) = 0, for all t t t y ∈ X − x. We use the notation x ∈ μ if and only if x ∈ μ . t t Definition 2.6 ([7]): Let Y be a subset of a non-empty set X and t ∈ (0, 1], then t is defined as t,if x ∈ Y t (x) = 0, otherwise When t = 1 then 1 , is known as the characteristic function of Y and it is denoted by χ . Y Y Definition 2.7 ([8]): A fuzzy left ideal μ of R is called a essential fuzzy left ideal of R, denoted by μ ⊆ R, if for every nonzero fuzzy left ideal δ of R, there exist x(= 0) ∈ R such that x ∈ μ e t and x ∈ δ , for some t ∈ (0, 1]. Definition 2.8 ([8]): Let μ and σ be two nonzero fuzzy left ideals of R such that μ ⊆ σ . Then μ is called a fuzzy essential left ideal in σ , denoted by μ ⊆ σ if for every nonzero e 214 U. MEDHI ET AL. fuzzy left ideal δ of R satisfying δ ⊆ σ , there exist x(= 0) ∈ R such that x ∈ μ and x ∈ δ,for t t some t ∈ (0, 1]. Lemma 2.9 ([8]): A fuzzy left ideal μ of R is a essential fuzzy left ideal of R if and only if μ is fuzzy essential left ideal of χ . Lemma 2.10 ([8]): Let μ, ν and σ be nonzero fuzzy left ideals of R such that μ ⊆ ν ⊆ σ . Then μ ⊆ σ if and only if μ ⊆ ν ⊆ σ . e e e Definition 2.11: Let μ be a fuzzy subset of a non empty set X. Then the support of μ, ∗ ∗ denoted by μ is defined as μ ={x ∈ X : μ(x)> 0}.If μ is a fuzzy submodule of M then μ is a submodule of M. Definition 2.12 ([7]): Let ζ be a fuzzy ideal and μ ∈ F(M). We define ζμ as follows: (ζ μ)(x) =∨{ζ(r) ∧ μ(y) | r ∈ R, y ∈ M, ry = x}, ∀ x ∈ M. Definition 2.13 ([9]): Let μ be a fuzzy subset of an R-module M. Then the fuzzy subset ann(μ) of R is defined as follows: ann(μ) = {η | η ∈ [0, 1] , ημ ⊆ χ }. Lemma 2.14 ([9]): Let μ ∈ [0, 1] . Then ann(μ) = {r | r ∈ R, α ∈ [0, 1], r μ ⊆ χ }. α α 0 Definition 2.15 ([7]): Let μ be a fuzzy submodule of M and ν be any fuzzy subset of M. Then (μ : ν) = {η | η ∈ [0, 1] , ην ⊆ μ}. Lemma 2.16 ([7]): Let μ, ν be any two fuzzy subsets of M. Then μ : ν = {r | r ∈ R, α ∈ [0, 1], r ν ⊆ ν}. α α Definition 2.17: Let f be a R-module homomorphism from M to M , μ ∈ F(M) and ν ∈ −1 F(M ). Then f (μ) ∈ F(M ) and f (ν) ∈ F(M) are defined by 1 1 −1 sup −1 μ(m),if f (w) = φ m∈f (w) f (μ)(w) = 0, otherwise −1 and f (ν)(m) = ν(f (m)), for all w ∈ M , m ∈ M. 3. Fuzzy Submodules In this section let F be a non empty set of fuzzy left ideals of R. FUZZY INFORMATION AND ENGINEERING 215 Definition 3.1: Let M be a left R-module and μ be a fuzzy submodule of M. Then we define fuzzy torsion of μ as follows: T(μ) = {γ | γ ∈ [0, 1] , γ ⊆ μ, γσ ⊆ χ , for some fuzzy ideal σ }. Also we define the F-torsion of μ as follows: M M T (μ) = {γ | γ ∈ [0, 1] , γ ⊆ μ there is σ ∈ F such that γσ ⊆ χ } M M 1 is F-torsion if T (1 ) = 1 and F-torsion free if T (1 ) = χ . M M M M θ F F Definition 3.2: Let M be a left R-module and μ be a fuzzy submodule of M. Then we define fuzzy closure of μ as follows: Cl(μ) = {σ | σ ∈ [0, 1] , σ ⊆ μ, γσ ⊆ μ, for some fuzzy ideal γ }. Also we define the F-closure of μ as follows: M M Cl (μ) = {σ | σ ∈ [0, 1] , σ ⊆ μ, there is γ ∈ F such that γσ ⊆ μ}. Lemma 3.3: T(μ) = {m | m ∈ μ, m σ ⊆ χ , for some fuzzy ideal σ } α α α θ Proof: Clearly, {m | m ∈ μ, m σ ⊆ χ , for some fuzzy ideal σ } α α α θ ⊆{γ | γ ∈ [0, 1] , γ ⊆ μ, γσ ⊆ χ , for some fuzzy ideal σ }. Therefore, {m ∈ μ, m σ ⊆ χ , for some fuzzy ideal σ } α α θ ⊆ {γ | γ ∈ [0, 1] , γ ⊆ μ, γσ ⊆ χ , for some fuzzy ideal σ }. = T(μ) Let γ ∈ [0, 1] such that γσ ⊆ χ , for some fuzzy ideal σ.Let m ∈ m such that γ(m) = α. Now,(m σ)(x) =∨{m (s) ∧ σ(y) | x = sy; s ∈ M, y ∈ R} α α =∨{γ(m) ∧ σ(y) | x = my; s ∈ M, y ∈ R} ≤{γ(s) ∧ σ(y) | x = sy; s ∈ M, y ∈ R} = (γ σ )(x) ⊆ χ (x). Thus (m σ) ⊆ χ . α θ Therefore, T(μ) ⊆ {m | m ∈ μ, m σ ⊆ χ , for some fuzzy ideal σ }. α α α θ Hence the result follows. The proofs of the following lemmas are similar. Lemma 3.4: T (μ) = {m | m ∈ μ there is γ ∈ F such that m γ ⊆ χ } α α α θ Lemma 3.5: Cl(μ) = {m | m ∈ μ, γ m ⊆ μ, for some fuzzy ideal γ } α α α Lemma 3.6: Cl (μ) = {m | m ∈ μ, there is γ ∈ F such that γ m ⊆ μ} α α α Definition 3.7: Let M be a left R-module and μ ∈ F(M).Wecall μ is F-closed if Cl (μ) = μ Definition 3.8: F is called weak closed under intersection if for any μ , μ ∈ F there exists 1 2 σ ∈ F such that σ ⊆ μ ∩ μ . 1 2 216 U. MEDHI ET AL. Definition 3.9: F is called inductive if for any μ ∈ F and any left ideal σ ⊇ μ,wehave σ ∈ F. Definition 3.10: F is called left closed if for any r ∈ 1 and any μ ∈ F, there is σ ∈ F such t R that σ r ⊆ μ i.e. (μ : r ) ⊆ σ . t t Definition 3.11: F is called a topological filter if it is closed under intersection, inductive and left closed. Theorem 3.12: If F is the set of all fuzzy essential ideals of R, then F is inductive. Proof: Let μ ∈ F.If σ is any fuzzy left ideal of R such that σ ⊇ μ. Then μ ⊆ R and so μ ⊆ e e 1 .Thusfrom μ ⊆ σ ⊆ 1 , it follows that σ ⊆ R and hence σ ∈ F. R R e Theorem 3.13: Let M be a left R-module. (a) If F is weak closed under intersection, then for any μ ∈ F(M) we have that Cl (μ) is a fuzzy submodule of M. M M M (b) If F is weak closed under intersection if and only if Cl (σ ) ∩ Cl (σ ) = Cl (σ ∩ σ ),for 1 2 1 2 F F F any two σ , σ ∈ F(M). 1 2 (c) If F is weak closed under intersection, then F is left closed if and only if Cl (σ ) is a fuzzy submodule of M for any σ ∈ F(M). Proof: (a) Let m , m ∈ M. 1 2 M M Now, Cl (μ)(m ) ∧ Cl (μ)(m ) 1 2 F F = ( {γ (m ) | γ ∈ [0, 1] , γ ⊆ μ, there is μ ∈ F such that μ γ ⊆ μ}) 1 1 1 1 1 1 1 ( {γ (m ) | γ ∈ [0, 1] , γ ⊆ μ, there is μ ∈ F such that μ γ ⊆ μ}) 2 2 2 1 2 2 2 = {γ (m ) ∧ γ (m ) | γ ∈ [0, 1] , γ ⊆ μ, there is 1 1 2 2 i i μ ∈ F such that μ γ ⊆ μ; i = 1, 2} i i i ≤{(γ + γ )(m ) ∧ (γ + γ )(m ) | γ ∈ [0, 1] , γ ⊆ μ, there is 1 2 1 1 2 2 i i μ ∈ F such that μ γ ⊆ μ; i = 1, 2} i i i ≤{(γ + γ )(m − m ) | μ(γ + γ )} 1 2 1 2 1 2 Since F is weak closed, it follows that μ , μ ∈ F implies there is σ ∈ F such that σ ⊆ 1 2 μ ∩ μ . Therefore we have, 1 2 σ(γ + γ ) ⊆ (μ ∩ μ )(γ + γ ) 1 2 1 2 1 2 ⊆ (μ ∩ μ )γ + (μ ∩ μ )γ 1 2 1 1 2 2 ⊆ μ γ + μ γ 1 1 2 2 ⊆ μ + μ = μ M M Thus, Cl (μ)(m ) ∧ Cl (μ)(m ) 1 2 F F ≤{(γ + γ )(m − m ) | σ(γ + γ )(m − m ) ⊆ μ, for some σ ∈ F} 1 2 1 2 1 2 1 2 = Cl (μ)(m − m ) 1 2 Also Cl (μ)(rm) = {γ(rm) | γ ∈ [0, 1] , γ ⊆ μ, there is σ ∈ F such that σγ ⊆ μ} FUZZY INFORMATION AND ENGINEERING 217 ≥ {γ(m) | γ ∈ [0, 1] , γ ⊆ μ, there is σ ∈ F such that σγ ⊆ μ} = Cl (μ)(m) Therefore Cl (μ) ∈ F(M). (b) (⇒). For any m ∈ M, Cl (σ ∩ σ )(m) 1 2 = {γ(m) | γ ∈ [0, 1] , γ ⊆ (σ ∩ σ ), there is σ ∈ F such that σγ ⊆ σ ∩ σ } 1 2 1 2 = {γ(m) ∧ γ(m) | γ ∈ [0, 1] , γ ⊆ (σ ∩ σ ), there is σ ∈ F such that 1 2 σγ ⊆ σ , σγ ⊆ σ } 1 2 = ( {γ(m) | γ ∈ [0, 1] , γ ⊆ σ , there is σ ∈ F such that σγ ⊆ σ }) 1 1 ( {γ(m) | γ ∈ [0, 1] , γ ⊆ σ , there is σ ∈ F such that σγ ⊆ σ }) 2 2 M M ≤ Cl (σ )(m) Cl (σ )(m) 1 2 F F M M = [Cl (σ ) ∩ Cl (σ )](m). 1 2 F F M M M Thus Cl (σ ∩ σ ) ⊆ Cl (σ ) ∩ Cl (σ ). 1 2 1 2 F F F M M Also, [Cl (σ ) ∩ Cl (σ )](m) 1 2 F F M M = Cl (σ )(m) Cl (σ )(m) 1 2 F F = ( {γ (m) | γ ∈ [0, 1] , γ ⊆ σ , there is μ ∈ F such that μ γ ⊆ σ }) 1 1 1 1 1 1 1 1 ( {γ (m) | γ ∈ [0, 1] , γ ⊆ σ , there is μ ∈ F such that μ γ ⊆ σ }) 2 2 2 2 2 2 2 2 = {γ (m) ∧ γ (m) | γ ∈ [0, 1] , γ ⊆ σ , there is μ ∈ F such that 1 2 i i i i μ γ ⊆ σ ; i = 1, 2} i i i = {(γ ∩ γ )(m) | γ ∈ [0, 1] , γ ⊆ σ , there is μ ∈ F such that 1 2 i i i i μ γ ∩ μ γ ⊆ σ ∩ σ ; i = 1, 2} Now, we have 1 1 2 2 1 2 (μ ∩ μ )(γ ∩ γ ) ⊆ (μ ∩ μ )γ ∩ (μ ∩ μ )γ 1 2 1 2 1 2 1 1 2 2 ⊆ γ μ ∩ γ μ 1 1 2 2 ⊆ σ ∩ σ 1 2 Since F is weak closed, for μ , μ ∈ F there exists μ ∈ F such that μ ⊆ μ ∩ μ . 1 2 1 2 Therefore, μ(γ ∩ γ ) ⊆ (μ ∩ μ )(γ ∩ γ ) ⊆ σ ∩ σ . 1 2 1 2 1 2 1 2 M M Thus we have, [Cl (σ ) ∩ Cl (σ )](m) 1 2 F F ≤ {γ(m) | γ ∈ [0, 1] , γ ⊆ (σ ∩ σ ), there is μ ∈ F 1 2 such that μγ ⊆ σ ∩ σ } 1 2 = Cl (σ ∩ σ ). Hence the result follows. 1 2 (⇐).Let μ , μ ∈ F. 1 2 We have, (μ 1 )(x) =∨{μ (y) ∧ 1 (z) | yz = x, y ∈ R}≤ μ (x) ∧ α ≤ μ (x). i α i α i i R R R Thus μ 1 ⊆ μ , ∀i = 1, 2 and therefore 1 ∈ [Cl (σ ) ∩ Cl (σ )] = Cl (σ ∩ σ ), hence i α i α 1 2 1 2 F F F there exists μ ∈ F such that μ ⊆ σ ∩ σ . 1 2 (c) (⇒). Follows from part (a). R R (⇐).Let μ ∈ F and r ∈ 1 , then Cl (μ) = 1 , hence r ∈ Cl (μ) and therefore there is t R R t F F ν ∈ F such that νr ⊆ μ. Theorem 3.14: Let F be an inductive set of fuzzy ideals, then the following statements are equivalent: (a) F is a topological filter. (b) Cl (σ ) is a fuzzy submodule for any σ ∈ F(M). F 218 U. MEDHI ET AL. Proof: (a) ⇒ (b). F closed under intersection, so weak closed under intersection. It is given that F is left closed, hence by part (c) of the above theorem the result follows. (b) ⇒ (a). F is weak closed under intersection and left closed as Cl (σ ) is a fuzzy sub- module for any σ ∈ F(M).Since F is inductive, therefore it is closed under intersection. Hence F is a topological filter. Definition 3.15: Let M be an R-module. Then a fuzzy submodule μ of M is said to F-closed in 1 if for any fuzzy ideal σ ∈ F and any m ∈ 1 , χ = σ m ⊆ μ implies m ∈ μ, α ∈ M α M θ α α (0, 1). Theorem 3.16: Let μ ⊆ σ ⊆ 1 .If μ is F-closed in σ , σ is F-closed in 1 then μ is F-closed in M M 1 . Proof: Let δ ∈ F and m ∈ 1 such that χ = m δ ⊆ μ.Thus m δ ⊆ μ ⊆ σ.Since σ is F- α M θ α α closed in 1 , it follows that m ∈ σ . Now, m ∈ σ and m δ ⊆ μ.Since μ is F-closed in σ,so M α α α it gives m ∈ μ. Hence μ is F-closedin1 . α M Theorem 3.17: Let σ ∈ F(M).If σ is F-closed in 1 and inductive then, for any m ∈ M α Cl (σ )\σ , we have (σ : m ) = Ann(m ). α α Proof: Suppose σ is F-closedin1 and inductive. Clearly Ann(m ) ⊆ (σ : m ). M α α Let m ∈ Cl (σ )\σ . Then there exists μ ∈ F such that μm ⊆ χ and this implies μ ⊆ α α θ Annm .Since F isinductive,sowehave Ann(m ) ∈ F.Now Ann(m ) ⊆ (σ : m ) implies α α α α (σ : m ) ∈ F. Next let r ∈ (σ : m ), then r m ⊆ σ , therefore (σ : m )m ⊆ σ and thus (σ : α t α t α α α m )m ⊆ χ . This implies (σ : m ) ⊆ Ann(m ). Hence (σ : m ) = Ann(m ). α α θ α α α α Theorem 3.18: Let σ ∈ F(M). For any m ∈ Cl (σ )\σ , we have (σ : m ) = Ann(m ), then α α α 1 m ∩ σ = χ , R α θ Proof: We have 1 m (x) =∨{1 (y) ∧ m (z) | y ∈ R, z ∈ M, x = yz} R α R α =∨{m (z) | y ∈ R, z ∈ M, x = yz} 0if x ∈ / Rm α if x ∈ Rm = (Rm) ∗ ∗ Then (Rm) is a fuzzy submodule. Let μ = (σ : m ) and L ={r ∈ R | rm ∈ σ }.Let x ∈ μ , α α then ∃α ∈ [0, 1] such that (xm) ∈ σ and therefore σ(xm) ≥ α ∧ p > 0. Consequently, α∧p ∗ ∗ xm ∈ σ and thus x ∈ L. Hence μ ⊆ L Again let x ∈ L, then x ∈ σ and this implies σ(xm)> 0. Let σ(xm) = α, then σ(xm)> α ∧ p,which gives (xm) ∈ σ.Thus x ∈ μ i.e. μ(x) ≥ α> 0 therefore x ∈ μ ,thus α∧p α ∗ ∗ ∗ ∗ ∗ ∗ L ⊆ μ . Hence μ = L.Thus (σ : m ) = L ={r | rm ∈ σ }= (σ : m).Also[Ann(m )] = α α ∗ ∗ Ann(m).Now,[(Rm) ∩ σ ] ={x | (Rm) ∧ σ(x)> 0}={x | x ∈ Rm and x ∈ σ }= Rm ∩ α α ∗ ∗ ∗ σ .By hypothesis we have, (σ : m ) = Ann(m ),thisimplies (σ : m) = [Ann(m )] = α α α FUZZY INFORMATION AND ENGINEERING 219 ∗ ∗ Ann(m) and therefore Rm ∩ σ = 0. So, [(Rm) ∩ σ ] = 0. Hence (Rm) ∩ σ = 1 m ∩ α α R α σ = χ . Theorem 3.19: Let σ ∈ F(M). If for any m ∈ Cl (σ )\σ , we have 1 m ∩ σ = χ , thenμ is α R α θ F-closed. Proof: Let μ ∈ F, m ∈ 1 such that χ = μm ⊆ σ.If m ∈ / σ then we have, by hypoth- α M θ α α esis 1 m ∩ σ = χ .Now μm ⊆ σ implies μm ∩ σ = μm ⊆ 1 m . Therefore μm = R α θ α α α R α α μm ∩ σ ⊆ 1 m ∩ σ = χ , a contradiction. Hence the result follows. α R α θ As the consequences of the Theorems 3.17, 3.18, 3.19 we obtain the following: Theorem 3.20: Let σ ∈ F(M) and F be inductive. Then the following statements are equiva- lent: (a) σ is F-closed in 1 . (b) For any m ∈ Cl (σ )\σ , we have (σ : m ) = Ann(m ). α α α (c) For any m ∈ Cl (σ )\σ , we have 1 ∩ m = χ . α R α θ M M Theorem 3.21: Let σ ∈ F(M) be an F-closed, then Cl (σ ) = σ ∪ T (1 ). F F M M M Proof: Clearly σ ∪ T (1 ) ⊆ Cl (σ ). Now let m ∈ Cl (σ )\σ , there exists μ ∈ F such that M α F F F μm = χ ,thus m ∈ T (1 ). α θ α M Acknowledgments The authors are really thankful to the referee(s) for reading the article and for making valuable helpful comments. Disclosure statement No potential conflict of interest was reported by the author(s). Funding The authors did not receive any funding for this research. Notes on contributors Ujwal Medhi is an Assistant Professor in the Department of Mathematics at Arya Vidyapeeth College, Guwahati, India. He completed Ph.D. degree under the supervision of Prof. H. K. Saikia from Gauhati University, India. His area of interest is fuzzy algebra. Helen K Saikia is a Professor in the Department of Mathematics at Gauhati University, Guwahati, India. Her areas of research interest are number theory, algebra, fuzzy algebra and graph theory. Sixteen research scholars have obtained Ph.D. degree under direct supervision of her. She has number of publications in reputed international journals. Bijan Davvaz is a full professor in the Department of Mathematics at Yazd University, Yazd, Iran. He works on algebra, algebraic hyperstructures, rough sets and fuzzy logic. He is a member of editorial 220 U. MEDHI ET AL. boards for 25 mathematical journals. He has authored around 500 research papers, especially on alge- bra, algebraic hyperstructures and their applications and fuzzy logic. He has also published five books on algebra. References [1] Abian A, Reinhart D. Honest subgroups of abelian groups. Rend Circ Math Palermo. 1963;12:353–356. [2] Fay TH, Joubert SV. Isolated submodules and skew fields. Appl Categorial Struct. 2000;8:317–326. [3] Pascual J. Honest submodules. Czochoslovak Math J. 2007;57(132):225–241. [4] Jouvert SV, Schoeman MJ. Superhonesty for modules and abelian groups. J Math. 1984;12(2):87– [5] Cheng MH. Notes on superhonest submodules. Chinese J Math. 1986;14(2):109–119. [6] Zadeh LA. Fuzzy sets. Inf Control. 1965;8:338–353. [7] Mordeson JN, Malik DS. Fuzzy commutative algebra. Singapore: World Scientific; 1998. [8] Medhi U, Rajkhowa KK, Barthakur LK, et al. On fuzzy essential ideals of rings. Adv Fuzzy Sets Syst. 2008;3(3):287–299. [9] Kalita MC, Saikia HK. On annihilators of fuzzy subsets of modules. Int J Algebra. 2009;3(10):483– [10] Liu W. Fuzzy invariant subgroups and fuzzy ideals. Fuzzy Sets Syst. 1982;8:133–139.

Journal

Fuzzy Information and EngineeringTaylor & Francis

Published: Apr 3, 2019

Keywords: Fuzzy submodule; fuzzy closure; fuzzy torsion; 08A72; 16D10

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