Fuzzy Soc-Semi-Prime Sub-Modules

In this paper, we study a new concept of fuzzy sub-module, called fuzzy socle semi-prime sub-module that is a generalization the concept of semi-prime fuzzy sub-module and fuzzy of approximately semi-prime sub-module in the ordinary sense. This leads us to introduce level property which studies the relation between the ordinary and fuzzy sense of approximately semi-prime sub-module. Also, some of its characteristics and notions such as the intersection, image and external direct sum of fuzzy socle semi-prime sub-modules are introduced. Furthermore, the relation between the fuzzy socle semi-prime sub-module and other types of fuzzy sub-module presented. Keyword: F-module, F-sub-module, F-prime sub-module, Socle of F-module. 1.Introduction The concept of fuzzy sets was introduced by Zadeh in1965[1]. Many authors indeed presented fuzzy subrings and fuzzy ideals. The concept of fuzzy module was introduced by Negoita and Relescu in 1975 [2]. Since then several authors have studied fuzzy modules. The concept of semi-prime fuzzy sub-module was introduced by Rabi 2004[3]. The concept of approximately semi-prime sub-module was introduced by Ali 2019[4]. The socle of M is a summation of simple sub-modules of an R-module M and denoted by Soc(M). But, the fuzzy socle of F-module X an R-module M is a summation of simple F-sub-modules of X and denoted by F − Soc(X). Ibn Al Haitham Journal for Pure and Applied Science Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index Doi: 10.30526/35.1.2804 Article history: Received 1, November, 2021, Accepted,16 , December, 2021, Published in January 2022. Saad S.Merie SaadSaleem@uokirkuk.edu.iq Depatment of Mthmatics, College of Education of Pure Science, Ibn AlHaitham, University of Baghdad, Baghdad – Iraq. Hatam Yahya Khalf dr.hatamyahya@yahoo.com Depatment of Mthmatics, College of Education of Pure Science, Ibn AlHaitham, University of Baghdad, Baghdad – Iraq. Ibn Al-Haitham Jour. for Pure & Appl. Sci. 5 3 (1)2022 103 Preliminaries " There are various definitions and characteristics in this section of F-sets , F-modules , and prime F-sub-modules. Definition 1.1 [1] Let D be a nonempty set and I is closed interval [0, 1] of real numbers. An F-set B in D (an F-subset of D) is a function from D into I. Definition 1.2 [1] AN F-set B of a set D is said to be F-constant if B(x) = t, ∀ x ∈ D t ∈ [0, 1] Definition 1.3 [1] Let xt: D → [0, 1] be an F-set in D, where x ∈D , t ∈ [0, 1] defined by: xt(y) = { t if x = y 0 if x ≠ y for all y ∈ D. xt is said to be an F-singleton or F-point in D. Definition 1.4 [5] Let B be an F-set in D, for all t ∈ [0, 1], the set Bt = {x ∈ D; B(x) ≥ t} is said to be a level subset of B. Remark 1.5 [6] Let Α and Β be two F-sets in S, then: 1Α = Β if and only if Α(x) = Β(x) for all x ∈ S. 2Α ⊆ Β if and only if Α(x) ≤ Β(x) for all x ∈ S. 3Α = Β if and only if Αt = Βt for all t ∈ [0,1]. If Α < B and there exists x ∈S such that Α(x) < Β(x), then A is a proper F-subset of Β and written as Α < Β. By part (2), we can deduce that xt ⊆ Α if and only if Α(x) ≥ t . Definition 1.6 [6] If Μ is an R-module. An F-set X of Μ is called F-module of an R-module Μ if : 1X(x − y) ≥ min{X(x), X(y)}for all x, y ∈ Μ}. 2X(rx) ≥ X(x) for all x ∈ Μ and r ∈ R . 3X(0) = 1. Proposition 1.7 [7] Let C be an F-set of an R-module Μ. Then the level subset Ct of Μ , ∀ t ∈ [0, 1] is a submodule of M if and only if C is an F-sub-module of F-module of an R-module Μ. Definition 1.8 [8] Let X and A be two F-modules of R-module Μ. A is said to be an F-sub-module of X if Α ⊆ X. Proposition 1.9 [5] Ibn Al-Haitham Jour. for Pure & Appl. Sci. 5 3 (1)2022 104 Let Α be an F-set of an R-module Μ. Then the level subset Αt , t ∈ [0, 1] is a sub-module of Μ if Α is an F-sub-module of X where X is an F-module of an R-module Μ. Now, we go over various F-sub-module attributes that will be useful in the next section. Lemma 1.10 [6] If rt be an F-singleton of R and Α be an F-module of an R-module Μ.Then for any w ∈ Μ (rt Α)(w) = { sup{inf (t, A(x))}: if w = rx} for some x ∈ Μ 0 otherwise Where rt: R → [0, 1], defined by rt(z) = { t if r = z 0 if r ≠ z For all z ∈ R Definition 1.11 [6] Let Α and Β be two F-sub-modules of an F-module X of R-module Μ. The residual quotient of Α and Β denoted by (Α ∶ Β) is the F-subset of R defined by: (Α ∶ Β)(r) = sup {t ∈ [0, 1] ∶ rt Β ⊆ Α}, for all r ∈ R. That is (Α ∶ Β) = {rt ∶ rt B ⊆ Α; rt is an F − singleton of R}. If Β = 〈xk 〉, then (Α ∶ 〈xk 〉) = {rt ∶ rt xk ⊆ Α; rt is an F − singleton of R }. Lemma 1.12 [9] Let Α be an F-sub-module of F-module X, (Α t : Xt ) ≥ (Α: X)t ,For all t ∈ [0, 1]. Also , we can prove that by Lemma 2.3.3.[6]. It follows that if , X = Α ⊕ Β,where A, Β ≤ X then Xt = (Α ⊕ Β)t = Αt ⊕ Βt . Definition 1.13 [10] Let f be a mapping from a set Μ into a set Ν and let Α be F-set in Μ. The image of Α is denoted by f (Α), where f (Α) is defined by: f (Α) (y) = { sup{Α(z): z ∈ f(y) ≠ ∅} for all y ∈ Ν 0 otherwise Note that, if f is a bijective mapping, then f (Α)(y) = Α(f(y)) Proposition 1.14 [11] Let f be a mapping from a set Μ into a set Ν. Assume that X and Y are F-modules of M and N respectively, let Α be an F-sub-module of X, then f (Α) is an F-sub-module of Y. Definition 1.15 [12] An F-subset K of a ring R is called F-ideal of R, if ∀ x, y ∈ R : 1K(x − y) ≥ min {K(x), K(y)} . 2K(xy) ≥ max {K(x), K(y)} . Definition 1.16 [13] Let X be an F-module of an R-module Μ, let A be an F-sub-module of X and K be an F ideal of R, the product KA of K and Α is defined by: KΑ(x) = { sup {inf{K(r1 ), ... . , K(rn ), Α(x1 ), ... , Α(xn )}} for some ri ∈ R, xi ∈ Μ, n ∈ Ν 0 otherwise Ibn Al-Haitham Jour. for Pure & Appl. Sci. 5 3 (1)2022 105 Note that K Α is an F-sub-module of X, and (KΑ)t = Kt Αt ,∀ t ∈ [0, 1]. Definition 1.17 [9] Let X be an F-module of an R-module Μ, An F-sub-module U of X is called completely prime if whenever rbmt ⊆ U,with rb ≠ 01 is an F-singleton of R and mt is an Fsingleton of Ximplies that mt ⊆ U for each t, b ∈ [0,1]. Definition 1.18 [6] Let Α and Β be two F-sub-modules of an R-module Μ. The addition A + Β is defined by: (Α + Β)(x) = sup{min{Α(y), Β(z)} with x = y + z, for all x, y, z ∈ Μ }. Furthermore, Α + Β is an F-sub-module of an R-module Μ. Corollary 1.19 [8] If X is an F-module of an R-module Μ and xt ⊆ X, then for all F-singleton rk of R, rk xt = (rx)λ, where λ = min {t, k}. Proposition 1.20 [6] Let Α and Β be two F-sub-modules of an F-module X of an R-module Μ. Then the residual quotient of Α and Β (Α ∶ Β) is an F-ideal of R. Proposition 1.21 [14] Let f: M ⟶ Ν be an R-homomorphisim, then f(Soc(M)) ⊆ Soc(Ν). Definition 1.22 [15] Let X be an F-module of an R-module Μ, X is called F-simple if and only if X has no proper F-sub-modules (in fact X is F-simple if and only if X has only itself and 01 ). Definition 1.23 [16] A F-module X is called semi-simple if X is a summation of simple F-sub-modules of X . Moreover, X is called semi-simple if X = F − Soc(X). Definition 1.24 [9] Let X be an F-module of an R-module Μ, X is said to be faithful if F − annX = 01 . Where F − annX = {rt ∶ rt xl = 01 ; for all xl ⊆ X and rt be an F − singleton of R}. Definition 1.25 [17] Let X be an F-module of an R-module Μ, X is said to be cancellative if whenever rt xl = rt yd for all xl , yd ⊆ X and rt be an F − singleton of R then xl = yd . Definition 1.26 [3] A proper F-sub-module U of an F-module X of an R-module M is called semi-prime Fsub-module of X if whenever rb mt ⊆ U,where rb is an F-singleton of R , mt is an Fsingleton of X and n ∈ Zimplies that rbmt ⊆ U for each t, b ∈ [0,1]. Definition 1.27 [4] A proper sub-module E of an R-module Μ is called pproximately semi prime (for a short app-semi-prime) sub-module of Μ if whenever am ∈ E, for a ∈ R, m ∈ Μimplies that am ∈ E + Soc(Μ) . Ibn Al-Haitham Jour. for Pure & Appl. Sci. 5 3 (1)2022 106 Definition 1.28 [9] An F-sub-module N of an F-module X of an R-module M is called weakly pure F-submodule of X if for any F-singleton rb of R implies that rbN = rbX ∩ N with b ∈ [0,1]. Lemma 1.29 [18] Let X be an F-module of an R-module M and let Α , Β and C are F-sub-modules of X such that C ⊆ Β. Then C + (Β⋂Α) = (C + Α)⋂Β. Proposition 1.30 [14] If Μ be a faithful multiplication R-module, then Soc(R)Μ = Soc(Μ) Definition 1.31 [15] Let X be an F-module of an Rmodule Μ. X is called multiplication F-module if and only if for each Fsub-module Α of X ,there exists an F-ideal K of R such that Α = KX. Proposition 1.32 [15] AN F-module X of an R-module Μ is a multiplication if and only if every non-empty F sub-module A of X such that Α = (Α:R X)X . Definition 1.33 [19] A sub-module V of R-module Μ is called essential if H ∩ V ≠ 0. For non-trivial sub-module H of Μ . Definition 1.34 [9] Let X be an F-module of an R-module Μ. An F-sub-module A of X is called essential if A ∩ B ≠ 01 , for nontrivial F-sub-module B of X. Finally, (shortly fuzzy set, fuzzy sub-module, fuzzy ideal, fuzzy module and fuzzy singleton are F-set, F-sub-module, F-ideal , F-module and F-singleton)." F-Soc-semi-prime sub-modules In this section, we offer the concept of an F-Soc-semi-prime sub-module as a generalization of ordinary concept(approximately semi-prime sub-module). Some characterizations of FSoc-prime sub-module are introduced. Definition 2.1 Let rb be an F-singleton of R and mt is an F-singleton of X , then a proper F-sub-module U of an F-module X of an R-module M is called an F-Socle semi-prime ( for short F-Socsemi-prime) sub-module(ideal) of X if whenever rb mt ⊆ U with n ∈ Z + implies that rbmt ⊆ U + F − Soc(X) for each t, b ∈ [0,1]. Furthermore, if rband sh are F-singletons of R, then a proper F-ideal L of R is called an F-Socle semi-prime ( for short F-Soc-semi-prime) ideal of R if whenever rb sh ⊆ L with n ∈ Zimplies that rbsh ⊆ L + F − Soc(R) for each h, b ∈ [0,1]. We will adopt the definition of an F-socle of X in this research as follows: Ibn Al-Haitham Jour. for Pure & Appl. Sci. 5 3 (1)2022 107 F − Soc(X): M → [0,1] such that: F − Soc(X)(m) = { 1 if m ∈ Soc(M) h if m ∉ Soc(M) with 0 < h < 1 Lemma 2.2 (F − Soc(X))t = Soc(Xt) for any F-module X for each t ∈ (0,1] with (F − Soc(X))t ≠ Xt Proof: F − Soc(X): M → [0,1] such that: F − Soc(X)(m) = { 1 if m ∈ Soc(M) h if m ∉ Soc(M) with 0 < h < 1 Now, (F − Soc(X))t = {m ∈ M ∶ (F − Soc(X))(m) ≥ t} So, if t = 1 then (F − Soc(X))t = Soc(M) = Soc(Xt) If 0 < t ≤ h then (F − Soc(X))t = M = Xt that is a contradiction If h < t < 1 then (F − Soc(X))t = Soc(M) = Soc(Xt) Lemma 2.3 Let X be an F-module of an R-module M with X(m)=1 for each m ∈ M, if U is an F-submodule of X is defined by U: M → [0,1] such that: U(m) = { 1 if m ∈ E k if m ∉ E with 0 < k < 1 Where E is a sub-module of M. Then U is an F-Soc-semi-prime sub-module of X if and only if E is an app-semi-prime sub-module of M. Proof: First of all, we must define U + F − Soc(X). (U + F − Soc(X))(m) = sup {min(U(y) , F − Soc(X)(z)) , y + z = m}


Definition 1.16 [13]
Let X be an ℱ-module of an ℛ-module Μ, let A be an ℱ-sub-module of X and K be an ℱideal of ℛ, the product KA of K and Α is defined by: Note that K Α is an ℱ-sub-module of X, and (KΑ) = K Α ,∀ t ∈ [0, 1].

Definition 1.17 [9]
Let X be an ℱ-module of an ℛ-module Μ, An ℱ-sub-module U of X is called completely prime if whenever ⊆ ,with ≠ 0 1 is an ℱ-singleton of ℛ and is an ℱsingleton of Ximplies that ⊆ for each t, b ∈ [0,1].

Definition 1.18 [6]
Let Α and Β be two ℱ-sub-modules of an R-module Μ. The addition A + Β is defined by:

Definition 1.22 [15]
Let X be an ℱ-module of an R-module Μ, X is called ℱ-simple if and only if X has no proper ℱ-sub-modules (in fact X is ℱ-simple if and only if X has only itself and 0 1 ).

Definition 1.23 [16]
ℱ-module is called semi-simple if is a summation of simple ℱ-sub-modules of . Moreover, is called semi-simple if = − ( ).

Definition 1.25 [17]
Let X be an ℱ-module of an ℛ-module Μ, X is said to be cancellative if whenever = for all , ⊆ X and be an ℱ − singleton of ℛ then = .

Definition 1.26 [3]
A proper ℱ-sub-module U of an ℱ-module X of an ℛ-module M is called semi-prime ℱsub-module of X if whenever ⊆ ,where is an ℱ-singleton of ℛ , is an ℱsingleton of X and n ∈ + implies that ⊆ for each t, b ∈ [0,1].

Definition 1.28 [9]
An ℱ-sub-module N of an ℱ-module X of an ℛ-module M is called weakly pure ℱ-submodule of X if for any ℱ-singleton of ℛ implies that = ∩ with b ∈ [0,1].

Definition 1.31 [15]
Let X be an ℱ-module of an ℛmodule Μ. X is called multiplication ℱ-module if and only if for each ℱsub-module Α of X ,there exists an ℱ-ideal K of ℛ such that Α = KX.

Proposition 1.32 [15]
AN ℱ-module X of an ℛ-module Μ is a multiplication if and only if every non-empty ℱsub-module A of X such that Α = (Α: ) .

Definition 1.33 [19]
A sub-module of ℛ-module Μ is called essential if ∩ V ≠ 0. For non-trivial sub-module H of Μ .

Definition 1.34 [9]
Let X be an ℱ-module of an ℛ-module Μ. An ℱ-sub-module A of X is called essential if ∩ ≠ 0 1 , for nontrivial ℱ-sub-module B of X.
Finally, (shortly fuzzy set, fuzzy sub-module, fuzzy ideal, fuzzy module and fuzzy singleton are ℱ-set, ℱ-sub-module, ℱ-ideal , ℱ-module and ℱ-singleton)." -Soc-semi-prime sub-modules In this section, we offer the concept of an ℱ-Soc-semi-prime sub-module as a generalization of ordinary concept(approximately semi-prime sub-module). Some characterizations of ℱ-Soc-prime sub-module are introduced.

Definition 2.1
Let be an ℱ-singleton of ℛ and is an ℱ-singleton of X , then a proper ℱ-sub-module U of an ℱ-module X of an ℛ-module M is called an ℱ-Socle semi-prime ( for short ℱ-Soc- Furthermore, if and ℎ are ℱ-singletons of ℛ, then a proper ℱ-ideal L of ℛ is called an ℱ-Socle semi-prime ( for short ℱ-Soc-semi-prime) ideal of ℛ if whenever ℎ ⊆ with n ∈ + implies that We will adopt the definition of an ℱ-socle of X in this research as follows: Where E is a sub-module of M. Then U is an ℱ-Soc-semi-prime sub-module of X if and only if E is an app-semi-prime sub-module of M. Proof: First of all, we must define + ℱ − ( ).
Suppose E is an app-semi-prime sub-module of M, to prove that U is an ℱ-Soc-semi-prime sub-module of X. Let ⊆ ℛ and That is mean is an app-semi-prime sub-module of .
Hence 1 = is an app-semi-prime sub-module of M.
The following example shows that the definition of an ℱ-socle of X that we adopt in this research is necessary to prove one side of above lemma.
We have is an app-semi-prime sub-module of M for every > 0 . Hence, U is not an ℱ-Soc-semi-prime of sub-module of X.

Proposition 2.5
Let U and V are ℱ-sub-modules of an ℱ-module X of an ℛ-module M with V is an ℱsemiprime sub-module of X. Then [ : ℛ ] is an ℱ-Soc-semi-prime ideal of ℛ.

Proposition 2.6
Let U and V are ℱ-Soc-semi-prime sub-modules of an ℱ-module X of an ℛ-module M with ℱ − ( ) ⊆ , Then U ∩V is an ℱ-Soc-semi-prime sub-module of X.

Remark 2.7
Every ℱ-semi-prime sub-module is an ℱ-Soc-semi-prime sub-module , but the converse is not true .
The following example show that the converse is not true Hence, U is not an ℱ-semi-prime sub-module of X.

Remark 2.9
Every completely ℱ-sub-module of an ℱ-module X of an ℛ-module M is an ℱ-Soc-semiprime sub-module of X, but the converse is not true .
The following example show that the converse is not true
2 is an app-semi-prime sub-module of M, so by (Lemma 2.3) we get U is an ℱ-Soc-semiprime sub-module of X.
But U is not completely ℱ-sub-module of X, since for an ℱ-singleton 51 3 ⊆ and an ℱ- Hence, U is not completely ℱ-sub-module of X.

Corollary 2.12
Let U be an ℱ-sub-module of an ℱ-module X of an ℛ-module M, Then U is an ℱ-Soc-semiprime sub-module of X if and only if ∀ ℱ-sub-module S of X and every ℱ-singleton of ℛ with ( ) S ⊆U implies that S ⊆ + ℱ − ( ) .

Proof:
It is clear from (proposition 2.11).

Corollary 2.13
Let L be an ℱideal of ℛ, Then L is an ℱ-Soc-semi-prime ideal of ℛ if and only if ∀ ℱ-subideal J of ℛ and every ℱ-singleton of ℛ with ( ) J ⊆L implies that J ⊆ + ℱ − (ℛ) .

Remark 2.15
Every ℱ-semi-prime sub-module is an ℱ-Soc-semi-prime sub-module. Proof: It is Clear by definition of ℱ-semi-prime sub-module.

Proof:
Assume that U is an ℱ-Soc-semi-prime sub-module of an ℱ-module X of an ℛ-module M.

Theorem 2.20
Any ℱ-sub-module of semi-simple ℱ-module X is an ℱ-Soc-semi-prime sub-module of X. Proof: 2) if ⨁ is an ℱ-Soc-semi-prime sub-module of ⨁ thus V is an ℱ-Soc-semi-prime sub-module of X.

Lemma 2.24 :
If X is an ℱ-module of an ℛ-module M, and M be a faithful multiplication ℛ-module, then:

Proposition 2.25 :
Let X be a finitely generated multiplication and faithful ℱ-module of an ℛ-module M, if J is an ℱ-Soc-semi-prime ideal of ℛ then JX is an ℱ-Soc-semi-prime sub-module of X.

Proposition 2.26
Suppose that U is an ℱ-Soc-semi-prime sub-module of an ℱ-module X and V is an ℱ-semiprime sub-module of X with ℱ − ( ) ⊆ . Then the intersection of U and V is an ℱ-Soc-semi-prime of X.

Proposition 2.27
Let X be a faithful multiplication ℱ-module of an ℛ-module M, then a proper ℱ-sub-module U is an ℱ-Soc-semi-prime sub-module of if and only if [ : ] is an ℱ-Soc-semi-prime ideal of ℛ.

2.Conclusion
Through this research, we were able to know some of the fuzzy algebraic properties of fuzzy socle semi-prime sub-modules and the relationship with other concepts .