Weak Essential Fuzzy Submodules Of Fuzzy Modules

Throughout this paper, we introduce the notion of weak essential F-submodules of Fmodules as a generalization of weak essential submodules. Also, we study the homomorphic image and inverse image of weak essential F-submodules.


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 ∶ ⟶ I, Definition 1.2 [2] Let x ∶ ⟶ I, be a F-set in S, where x ∈ S , t ∈ I, defined by: If x = 0 and t = 1 then : We shall call such F-singleton the F-zero singleton. Proposition 1.3 [3]: Let , be two F-singletons of a set S. If = , then a = b and t = k, where t, k ∈ I. Definition 1.4 [5]: Let , are F-sets in S, then : 1.

2.
⊆ 2 if and only if (x) x , ∀ x ∈ . If ⊂ and there exists x ∈ S such that (x) (x), then is called a proper Fsubset of .

x ⊆ A if and only x
A , ∀ y ∈ if t 0 ℎ A x t. Thus x ⊆ A (x ∈ ), ( that is x ∈ if and only if x ⊆ A) Definition 1.5 [5]: Let , are F-sets in S, then: 1.

2.
∩ x min x , x , ∀ x ∈ . ∪ ∩ are F-sets in S. In general if , ∈ Λ , is a family of F-sets in S, then: 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 ∈ , Α is called level . " The following are some properties of the level subset: Remark 1.7 [1]: Let Α, Β are F-subsets of S, t ∈ I, then:

A = B if and only if
, for all t [0,1].

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: A said F-set X is F-module of an ℛ-module ℳ if: 1. X( ) min X , X , ∀ , ∈ ℳ. 2. X(r ) X( ), ∀ ∈ 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 : ⟶ be Fhomomorphism. If and are two F-submodules of X and X resp., then: 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 , 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.  x ∈ ℳ: x 0 is called support of A, also  ∪ , t ∈ 0,1 .

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]: 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 Fsubmodule if and only if for each non-trivial semiprime F-submodule S of X, there exists x ⊆ and r of ℛ, such that x r ⊆ , ∀ ∈ 0,1 .

Proof:
Suppose that non-trivial semiprime F-submodule S of X, there exists x ⊆ and r of ℛ such that 0 x r ⊆ . Note that x r ⊆ . 0 x r ⊆ ∩ . Thus A∩ 0 , that is A is weak essential F-submodule. Conversely, A is weak essential F-submodule, then A∩ 0 , for each non-trivial semiprime F-submodule S of X. Thus, there exists 0 x ⊆ ∩ , implying that x ⊆ and hence 0 ⊆ , ∀ ∈ 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 weak essential submodule of X , ∀ ∈ 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 semiprime submodule of X , ∀ ∈ 0,1 , see [14,Theorem(2.4)], which implies ∩ 0 , since is weak essential submodule and ∩ ∩ 0 , hence A ∩ 0 by Remark (1.7)(3). Thus, A is a weak essential Fsubmodule 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 = as Z-module. Define X : ℳ ⟶ I, by: . Also is not weak essential, since ∩ 0 , where S any semiprime submodule. Therefore 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 * is weak essential submodule in X * . Proof: Let * 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 ∩ 0 ,then ∩ * 0 , implies that * ∩ * 0 . But S is semiprime F-submodule, then is semiprime see [14, Theorem (2.4)], so * is semiprime, hence * 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 * is weak essential submodule in X * . Let N is semiprime submodule of X * and * ∩ 0 , we must prove N = (0).
Define B : ℳ ⟶ I by: B(x) = 1 x ∈ 0 ℎ It is clear that B F-submodule of X, * , so * ∩ * 0 , then ∩ * 0 , hence by Remark(1.7)(3), A∩ 0 and B = 0 , since A is weak essential F-submodule in X, so * 0 ; therefore N = (0). Thus * is weak essential submodule in X * . Remarks 2.9: 1. Let Α, Β are F-submodules of X such that A ⊆ 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 . Let X : ℳ ⟶ I, define by : X(a) = 1, for all ∈ . Define A: ℳ ⟶ I , B : ℳ ⟶ I by: ℎ It is clear that X and A, B are F-submodules of X . a weak essential submodule in X see [4,Remarks(1.5)]. Thus B is weak essential Fsubmodule of X by Lemma (2.4). Let C : ℳ ⟶ I, as defined by: 2. Let A, B are F-submodule such that A ⊆ . If A is weak essential F-submodule in X implying Β is a weak essential F-submodule of X. Proof: Assume that B ∩ 0 , for some semi-prime F-submodule S of X, then A ∩ 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 ∩ 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 as Z-module. Define X : ℳ ⟶ I by: X(a) = 1, for all ∈ . Let A : ℳ ⟶ I, B : ℳ ⟶ I, define by: 18 , ∀ ∈ 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 ∩ 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 ∩ is a weak essential F-submodule of X. Proof: Suppose S is a non-trivial semiprime F-submodule of X, but B is weak essential Fsubmodule of X, hence B ∩ 0 . So A is an essential F-submodule of X and we have A ∩ ∩ A ∩ ∩ 0 , Hence, A ∩ B is weak essential F-submodule of X.
In the following proposition, we prove the transitive property for non-trivial Fsubmodule.

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 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 Fsubmodule 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 is a weak essential F-submodule of X such that is f-invariant, then f ( ) is a weak essential F-submodule of X . Proof: To show f ( ) is a weak essential F-submodule of X , since is a F-submodule of X , then f ( ) is a F-submodule of X by Proposition (1.13)(1).Now suppose that S semiprime Fsubmodule of X such that f ( ) ∩ 0 ; therefore (f ( )∩ 0 , then f ( ) ∩ 0 , see Proposition (1.10) (2). But is f-invariant implying that ∩ (S) 0 , and 0 , since is weak essential Ϝ-submodule and Fsubmodule of X by Proposition (1.13)(2). f ( 0 , then S = 0 , by Proposition (1.10) (3). That is f ( ) is a weak essential F-submodule. Now, we consider the inverse image of a weak F-submodule.

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 is weak essential F-submodule of X , then ( ) is a weak essential F-submodule of X . Proof: Since F-submodule of X , then is F-submodule of X see Proposition(1.13)(2).Now suppose S is semiprime F-submodule of X ,such that ∩ 0 , hence f ( ∩ 0 , implies that f ( ∩ 0 see Proposition (1.10)(6). ∩ 0 (since is f-invariant and f is epimorphism), then 0 ), implies that S = 0 , since every F-submodule of X is f-invariant, implies is weak essential F-submodule of X .