Microsoft Word - 65-72 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020   65          Weak Essential Fuzzy Submodules Of Fuzzy Modules Abstract Throughout this paper, we introduce the notion of weak essential F-submodules of F- modules as a generalization of weak essential submodules. Also, we study the homomorphic image and inverse image of weak essential F-submodules. Keywords: Semi-prime F-submodules, essential F-submodules. 1.Introduction Let S ∅. Zadeh [1] defined F-subset X of S as a mapping X: S ⟶[0,1]. Negoita and Ralescu [2] introduced the concept of F-modules. Mashinchi and Zahedi [3] introduced the notion of F-submodules. Mona [4] introduced and studied the concept of weak essential submodules, where a submodule Η of ℳ is called a weak essential, if H ∩ L (0), for each non-zero semiprime submodule L of ℳ. In this paper, we introduce the notion weak essential F- submodule of F-module. We investigate some basic results about weak essential submodules. Next, throughout this paper ℛ is a commutative ring with identity, ℳ is an ℛ-module and X is a F-module of an ℛ-module ℳ. Finally, (shortly fuzzy set, fuzzy submodule and fuzzy module is F-set, F-submodule and F- module). S.1 Preliminaries In this section, we shall give the concepts of F-sets and operations on F-sets, with some important properties of them, which are used in this paper. Ibn Al Haitham Journal for Pure and Applied Science Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index Hassan K. Marhon Ministry of Education, Rusafa1 hassanmath316@gmail.com Hatam Y. Khalaf Department of Mathematics, College of Education for pure Sciences, Ibn-Al- Haitham , Baghdad University, E-mail: dr.hatamyahya@yahoo.com Doi: 10.30526/33.4.2510 Article history: Received 27 November 2019, Accepted 16 December 2020, Published in October 2020   66  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 Definition 1.1 [1]: Let S be a non-empty set and let I be a closed interval [0,1] of the real line (real number ). A F-set X in S (a fuzzy subset X of S) is characterized by a membership function X ∶ 𝑆 ⟶ I, Definition 1.2 [2] Let x ∶ 𝑆 ⟶ I, be a F-set in S, where x ∈ S , t ∈ I, defined by: x 1 𝑖𝑓 𝑥 𝑦 0 𝑖𝑓 𝑥 𝑦 Then x a said F- singleton. If x = 0 and t = 1 then : 0 𝑦 1 𝑖𝑓 𝑦 0 0 𝑖𝑓 𝑦 0 We shall call such F-singleton the F-zero singleton. Proposition 1.3 [3]: Let 𝑎 , 𝑏 be two F-singletons of a set S. If 𝑎 = 𝑏 , then a = b and t = k, where t, k ∈ I. Definition 1.4 [5]: Let 𝐴 , 𝐴 are F-sets in S, then : 1. 𝐴 = 𝐴 if and only if 𝐴 (x) =𝐴 (x) , ∀ x ∈ 𝑆. 2. 𝐴 ⊆ 𝐴2 if and only if 𝐴 (x) 𝐴 x , ∀ x ∈ 𝑆. If 𝐴 ⊂ 𝐴 and there exists x ∈ S such that 𝐴 (x) 𝐴 (x), then 𝐴 is called a proper F- subset of 𝐴 . 3. x ⊆ A if and only x 𝑦 A 𝑦 , ∀ y ∈ 𝑆 𝑎𝑛𝑑 if t 0 𝑡ℎ𝑒𝑛 A x t. Thus x ⊆ A (x ∈ 𝐴 ), ( that is x ∈ 𝐴 if and only if x ⊆ A) Definition 1.5 [5]: Let 𝐴 , 𝐴 are F-sets in S, then: 1. 𝐴 ∪ 𝐴 (x) = max 𝐴 x , 𝐴 x , ∀ x ∈ 𝑆. 2. 𝐴 ∩ 𝐴 x min 𝐴 x , 𝐴 x , ∀ x ∈ 𝑆. 𝐴 ∪ 𝐴 𝑎𝑛𝑑 𝐴 ∩ 𝐴 are F-sets in S. In general if 𝐴 , 𝛼 ∈ Λ , is a family of F-sets in S, then: 𝐴 ∈ x 𝑖𝑛𝑓 𝐴 x , 𝛼 ∈ Λ , 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 ∈ 𝑆. 𝐴 ∈ x sup 𝐴 x , 𝛼 ∈ Λ , 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 ∈ 𝑆. Now, we give the definition of level subset, which is a set between F-set and ordinary set. Definition 1.6 [6]: Let Α be a F-set in S. For t ∈ I, the set 𝐴 𝑥 ∈ 𝑆, Α 𝑥 𝑡 is called level 𝒔𝒖𝒃𝒔𝒆𝒕 𝒐𝒇 𝐗 . " The following are some properties of the level subset: Remark 1.7 [1]: Let Α, Β are F-subsets of S, t ∈ I, then: 1. 𝐴 ∩ 𝐵 𝐴 ∩ 𝐵 . 2. 𝐴 ∪ 𝐵 𝐴 ∪ 𝐵 . 3. A = B if and only if 𝐴 𝐵 , for all t [0,1].   67  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 Definition1.8 [7]: Let f be a mapping from a set ℳ into a set ℳ , let A be a F-set in ℳ and B be a F-set in ℳ . The image of A denoted by f (A) is the F-set in ℳ defined by: f Α (y = 𝑠𝑢𝑝 𝐴 𝑧 | 𝑧 ∈ 𝑓 y 𝑖𝑓 𝑓 𝑦 ∅, 𝑓𝑜𝑟 𝑒𝑎𝑐ℎ y ∈ ℳ 0 𝑜. 𝑤 where 𝑓 y x ∶ 𝑓 x y And the inverse of B(x), denoted by 𝑓 B is the F-set in ℳ defined by: 𝑓 Β Β 𝑓 x , for all x ∈ ℳ . Definition 1.9 [8]: Let f be a function from a set ℳ into a set ℳ . A F-subset A of ℳ is a said f- invariant if A(x) = A(y), whenever f (x) = f (y), where x, y ∈ ℳ . Proposition 1.10 [8]: If f is a function defined on a set ℳ, 𝐴 𝑎𝑛𝑑𝐴 are F-subsets of ℳ, 𝐵 𝑎𝑛𝑑 𝐵 are F- subset of f (ℳ). The followings are true: 1. 𝐴 ⊆ 𝑓 𝑓 𝐴 . 2. 𝐴 𝑓 𝑓 𝐴 , whenever 𝐴 is f-invariant. 3. 𝑓 𝑓 𝐵 𝐵 . 4. If 𝐴 ⊆ 𝐴 , then 𝑓 𝐴 ⊆ 𝑓 𝐴 . 5. If 𝐵 ⊆ 𝐵 ,then 𝑓 𝐵 ⊆ 𝑓 𝐵 . 6. Let f be a function from a set ℳ into N. If 𝐵 and 𝐵 𝑎𝑟𝑒 F-subsets of N, then 𝑓 𝐵 ∩ 𝐵 =𝑓 𝐵 ∩ 𝑓 𝐵 [9] . Definition 1.11 [2]: A said F-set X is F-module of an ℛ-module ℳ if: 1. X(𝜈 𝜇) min X 𝜈 , X 𝜇 , ∀ 𝜈, 𝜇 ∈ ℳ. 2. X(r𝜈) X(𝜈), ∀ 𝜈 ∈ ℳ and r ∈ ℛ. 3. X(0) = 1 ( 0 is the zero element of ℳ ). Definition 1.12 [3]: Let X , X are F-modules of an ℛ-module ℳ. X is a said F-submodule of X if X ⊆ X ." Proposition 1.13 [10]: Let X , X be two F-modules of an ℛ-module ℳ and ℳ resp. Let f :𝑋 ⟶ 𝑋 be F- homomorphism. If 𝐴 and 𝐴 are two F-submodules of X and X resp., then: 1. 𝑓 𝐴 is a F-submodule of X . 2. 𝑓 𝐴 is a Ϝ-submodule of X . Proposition 1.14 [11]: Let Α be a F-set of an ℛ-module ℳ. Then, the level subset 𝐴 , t ∈ I, is a submodule of ℳ iff Α is Ϝ-submodule of X. Definition 1.15 [3]: Let A be a F-module in ℳ, then we define: 1. 𝐴 x ∈ ℳ: 𝐴 x 0 is called support of A, also 𝐴 ∪ 𝐴 , t ∈ 0,1 . 2. 𝐴 x ∈ ℳ: 𝐴 x 1 𝐴 0ℳ .   68  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 Definition1.16 [12]: A F-submodule Α of a F-module X is called an essential (briefly Α X), if Α ∩ 𝛣 0 , for any non-trivial F-submodule Β of X . 2. Weak Essential Fuzzy Submodules Mona in [4] introduced the concept of weak essential submodule, where a submodule Η of ℳ is a said weak essential, if H ∩ L (0), for each non-zero semiprime submodule L of ℳ, where a submodule N of an ℛ-module ℳ is called semiprime if for each r ∈ ℛ and m ∈ ℳ, if r x ∈ N, then rx ∈ N [13]. We shall fuzzify this concept. Definition 2.1 [14]: Let Α be F-submodule of F-module X is a said a semiprime Ϝ-submodule if 𝑟 𝑎 ⊆ 𝐴, for F-singleton 𝑟 of ℛ, 𝑎 ⊆ X, k ∈ 𝑍 , then 𝑟 𝑎 ⊆ 𝐴. Equivalently, A is semiprime F- submodule if 𝑟 ²𝑎 ⊆ 𝐴 for 𝑎 ⊆ X and 𝑟 a F-singleton of ℛ, then 𝑟 𝑎 ⊆ 𝐴. " Definition 2.2: Let 𝐴 be F-submodule of F-module X. 𝐴 is a said weak essential F-submodule if 𝐴 ∩ 𝑆 0 , for each non-trivial semiprime F-submodules of X. Equivalently F- submodule A of a F-module X is called weak essential F-submodule if A ∩ 𝑆 0 , then S 0 , for every semiprime F-submodule of X . Next, proposition is a characterization of a weak essential F-submodule. Proposition 2.3: Let X be a F-module and A a non-trivial F-submodule of X is a weak essential F- submodule if and only if for each non-trivial semiprime F-submodule S of X, there exists x ⊆ 𝑆 and r of ℛ, such that x r ⊆ 𝐴, ∀ 𝑡 ∈ 0,1 . Proof: Suppose that non-trivial semiprime F-submodule S of X, there exists x ⊆ 𝑆 and r of ℛ such that 0 x r ⊆ 𝐴. Note that x r ⊆ 𝑆. 0 x r ⊆ 𝐴 ∩ 𝐵. Thus A∩ 𝐵 0 , that is A is weak essential F-submodule. Conversely, A is weak essential F-submodule, then A∩ 𝑆 0 , for each non-trivial semiprime F-submodule S of X. Thus, there exists 0 x ⊆ 𝐴 ∩ 𝑆, implying that x ⊆ 𝐴 and hence 0 𝑥 𝑟 ⊆ 𝐴, ∀ 𝑡 ∈ 0,1 . Now, we give the following Lemma, which we will need in proving the next result. Lemma 2.4: Let A be a F-submodule of a F-module X if 𝐴 weak essential submodule of X , ∀ 𝑡 ∈ I. Then Α is weak essential F-submodule in X. Proof: Assume Β a semiprime F-submodule of X such that B 0 , since B semiprime F-submodule of X, hence 𝐵 semiprime submodule of X , ∀ 𝑡 ∈ 0,1 , see [14, Theorem(2.4)], which implies 𝐴 ∩ 𝐵 0 , since 𝐴 is weak essential submodule and 𝐴 ∩ 𝐵 𝐴 ∩ 𝐵 0 , hence A ∩ 𝐵 0 by Remark (1.7)(3). Thus, A is a weak essential F- submodule of X. Remark 2.5: Every essential F-submodule is weak essential F-submodule. But the converse is not true in general, for example: Example: Let ℳ = 𝑍 as Z-module. Define X : ℳ ⟶ I, by:   69  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 X(a) = 1, for all 𝑎 ∈ 𝑍 Let A : ℳ ⟶ I, define by: A(x) = 1 𝑖𝑓 x 0 1 2 𝑖𝑓 x ∈ 9 0 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 It is clear that A F-submodule of X, 𝐴 9 is weak essential by [4, Remarks(1.5)], then A is weak essential F-submodule by Lemma(2.4). Let B : ℳ ⟶ I, as defined by: B(x) = 1 𝑖𝑓 x 0 1 2 𝑖𝑓 x ∈ 4 0 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 It is clear that B F-submodule of X. A is not essential, since Α ∩ 𝐵 x 1 𝑖𝑓 x 0 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 A ∩ 𝐵 0 and B 0 ; therefore A is not essential F-submodule . Remark 2.6: The converse of Lemma (2.4) is not true in general. Example 2.7: Let ℳ = 𝑍 as Z-module. Define X : ℳ ⟶ I, A : ℳ ⟶ I by: X(a) = 1 𝑖𝑓 𝑎 0 1 2 𝑖𝑓 𝑎 2,4 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 , A(a) = 1 𝑖𝑓 𝑎 0 1 3 𝑖𝑓 𝑎 2,4 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 A is an essential F-submodule, then A is weak essential by Remark (2.5), but 𝐴 0 is not essential see [15, Remark (2.1)]. Also 𝐴 is not weak essential, since 𝐴 ∩ 𝑆 0 , where S any semiprime submodule. Therefore 𝐴 is not weak essential of X . Proposition 2.8: Let Α be a F-submodule of a F-module X, then Α is weak essential in X iff 𝐴∗ is weak essential submodule in X∗. Proof: Let 𝐴∗ is a weak essential submodule in X∗. To show A is weak essential F-submodule in X . Assume that S is semiprime F-submodule of X and A ∩ 𝑆 0 ,then 𝐴 ∩ 𝑆 ∗ 0 , implies that 𝐴∗ ∩ 𝑆∗ 0 . But S is semiprime F-submodule, then 𝑆 is semiprime see [14, Theorem (2.4)], so 𝑆∗ is semiprime, hence 𝑆∗ 0 , so S = 0 . Thus, A is weak essential F-submodule in X. Conversely, let A is a weak essential F-submodule in X, we have to show that 𝐴∗ is weak essential submodule in X∗. Let N is semiprime submodule of X∗ and 𝐴∗ ∩ 𝑁 0 , we must prove N = (0). Define B : ℳ ⟶ I by: B(x) = 1 𝑖𝑓 x ∈ 𝑁 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 It is clear that B F-submodule of X, 𝐵∗ 𝑁, so 𝐴∗ ∩ 𝐵∗ 0 , then 𝐴 ∩ 𝐵 ∗ 0 , hence by Remark(1.7)(3), A∩ 𝐵 0 and B = 0 , since A is weak essential F-submodule in X, so 𝐵∗ 0 ; therefore   70  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 N = (0). Thus 𝐴∗ is weak essential submodule in X∗. Remarks 2.9: 1. Let Α, Β are F-submodules of X such that A ⊆ 𝐵 and Β is weak essential F-submodule of X, then A need not be weak essential F-submodule for example: Let ℳ be as Z-module 𝑍 . Let X : ℳ ⟶ I, define by : X(a) = 1, for all 𝑎 ∈ 𝑍 . Define A: ℳ ⟶ I , B : ℳ ⟶ I by: A(x) = 1 𝑖𝑓 x ∈ 18 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 , B(x) = 1 𝑖𝑓 x ∈ 2 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 It is clear that X 𝑍 and A, B are F-submodules of X . 𝐵 a weak essential submodule in X see [4, Remarks(1.5)]. Thus B is weak essential F- submodule of X by Lemma (2.4). Let C : ℳ ⟶ I, as defined by: C(x) = 1 𝑖𝑓 x ∈ 12 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 , where C semiprime F-submodule C 12 , is semiprime submodule of X (∀ 𝑡 0 . But A ∩ 𝐶 0 , therefore Α is not weak essential F-submodule of X . 2. Let A, B are F-submodule such that A ⊆ 𝐵. If A is weak essential F-submodule in X implying Β is a weak essential F-submodule of X. Proof: Assume that B ∩ 𝑆 0 , for some semi-prime F-submodule S of X, then A ∩ 𝑆 0 . But A is weak essential F-submodule, hence S = 0 . That is B is weak essential F-submodule of X . 3. Let A, B be are F-submodules of F-module X if A ∩ 𝐵 a weak essential F-submodule of X, then both of A and B are weak essential F-submodules of X. Proof: It is clear by (2). Note that, the converse is not true in general, for example: Example: Let ℳ be 𝑍 as Z-module. Define X : ℳ ⟶ I by: X(a) = 1, for all 𝑎 ∈ 𝑍 . Let A : ℳ ⟶ I, B : ℳ ⟶ I, define by: A(x) = 1 𝑖𝑓 x ∈ 12 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 , B(x) = 1 𝑖𝑓 x ∈ 18 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 Clearly A, B are F-submodules of Χ, 𝐴 12 , 𝐵 18 , ∀ 𝑡 ∈ 0,1 are weak essential submodules of X by [4, Remark(1.5)]. Hence A, Β are weak essential F-submodules of X; see Lemma(2.4). But A ∩ 𝐵 0 ; that is A ∩ B is not weak essential F-submodule of X . Under some conditions the converse (3) will be true as in the following proposition. Proposition 2.10: Let A, B are F-submodules of F-module X such that A is an essential F-submodule, B weak essential F-submodule, then A ∩ 𝐵 is a weak essential F-submodule of X. Proof:   71  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 Suppose S is a non-trivial semiprime F-submodule of X, but B is weak essential F- submodule of X, hence B ∩ 𝑆 0 . So A is an essential F-submodule of X and we have A ∩ 𝐵 ∩ 𝑆 A ∩ 𝐵 ∩ 𝑆 0 , Hence, A ∩ B is weak essential F-submodule of X. Lemma 2.11: If S is a semiprime F-submodule of F-module X, Β be a F-submodule of X such that B ⊈ S, then S ∩ B is semiprime F-submodule in B. Proof: Let S be a semiprime F-submodule of X, then by [14,Theorem(2.4)], 𝑆 semiprime submodule and 𝐵 submodule of X ; see Proposition (1.14) such that 𝐵 ⊈ 𝑋 , then by [13 ,Proposition(1.11)], 𝑆 ∩ 𝐵 𝑆 ∩ 𝐵 ; see Proposition (1.7)(1) is a semiprime submodule in 𝐵 , therefore S ∩ B is a semiprime F-submodule in B; see [14, Theorem(2.4)]. In the following proposition, we prove the transitive property for non-trivial F- submodule. Proposition 2.12: Let A, B be a non-trivial F-submodules of F-module X such that A ⊆ B. If A is a weak essential F-submodule in B and B is a weak essential Ϝ-submodule in X implying A is a weak essential F-submodule in X. Proof: Assume that S is a semiprime F-submodule in X, such that A ∩ S =0 . Note that 0 A ∩ S = (A ∩ S) ∩ B = A ∩ (S ∩ Β). But S is a semi-prime F-submodule of X, so we have two cases. If B ⊆ S, then 0 A ∩ (S ∩ B) = A ∩ B. Hence, A ∩ B = 0 , but A ⊆ B so A ∩ B = A implies A = 0 which is a contradiction with our assumption. Thus B ⊈ S and by Lemma (2.11), S ∩ B is a semiprime F-submodule in B. Since A is a weak essential F- submodule in B, therefore S ∩ B = 0 and since B is a weak essential F-submodule in X, then S = 0 , then A is a weak essential F-submodule in X. Now, we study a homomorphic image of a weak essential F-submodule. Proposition 2.13: Let X , X be F-modules of an ℛ-module ℳ and ℳ resp. and f : X ⟶ X be F-epimorphism. If 𝐴 is a weak essential F-submodule of X such that 𝐴 is f-invariant, then f (𝐴 ) is a weak essential F-submodule of X . Proof: To show f (𝐴 ) is a weak essential F-submodule of X , since 𝐴 is a F-submodule of X , then f (𝐴 ) is a F-submodule of X by Proposition (1.13)(1).Now suppose that S semiprime F- submodule of X such that f (𝐴 ) ∩ 𝑆 0 ; therefore 𝑓 (f (𝐴 )∩ 𝑆 𝑓 0 , then 𝑓 f (𝐴 ) ∩ 𝑓 𝑆 0 , see Proposition (1.10)(2). But 𝐴 is f-invariant implying that 𝐴 ∩ 𝑓 (S) 0 , and 𝑓 𝑆 0 , since 𝐴 is weak essential Ϝ-submodule and 𝑓 𝑆 F- submodule of X by Proposition (1.13)(2). f (𝑓 𝑆 𝑓 0 , then S = 0 , by Proposition (1.10)(3). That is f (𝐴 ) is a weak essential F-submodule. Now, we consider the inverse image of a weak F-submodule.   72  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 Proposition 2.14: Let X , X are F-modules of an ℛ-module ℳ and ℳ resp. and f : X ⟶ X be F-epimorphism. If 𝐴 is weak essential F-submodule of X , then 𝑓 (𝐴 ) is a weak essential F-submodule of X . Proof: Since 𝐴 F-submodule of X , then 𝑓 𝐴 is F-submodule of X see Proposition(1.13)(2).Now suppose S is semiprime F-submodule of X ,such that 𝑓 𝐴 ∩ 𝑆 0 , hence f (𝑓 𝐴 ∩ 𝑆 𝑓 0 , implies that f (𝑓 𝐴 ∩ 𝑓 𝑆 𝑓 0 see Proposition (1.10)(6). 𝐴 ∩ 𝑓 𝑆 0 (since 𝐴 is f-invariant and f is epimorphism), then 𝑓 𝑓 𝑆 𝑓 0 ), implies that S = 0 , since every F-submodule of X is f-invariant, implies 𝑓 𝐴 is weak essential F-submodule of X . Reference 1. Zadeh, L.A. Fuzzy Sets. Information and Control. 1965, 8, 338-353. 2. Negoita, C. V.; Ralescu, D. A. Applications of fuzzy sets and System Analysis. (Birkhous Basel), 1975. 3. Mashinchi, M. ; Zahedi, M. M. On L-Fuzzy Primary Submodule. Fuzzy Sets and Systems. 1992, 49, 231-236. 4. Mona, A. A. weak Essential Submodules. Um-Salama, J. 2009, 6,1, 214-221. 5. Zahedi, M. M. On L-Fuzzy Residual Quotient Module and P. Primary Submodule. Fuzzy Sets and Systems. 1992, 51,333-344. 6. Martinez, L. 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