Neutrosophic Soft Modules

Neutrosophic Soft Modules Kemale Veliyeva, Sadi Bayramov Department of Algebra and Geometry, of Baku State University, 23,, Z.Khalilov str.,, AZ1148, Baku, Azerbaijan kemale2607@mail.ru, baysadi@gmail.com Abstract Molodtsov initiated the concept of soft sets in [17]. Maji et al. defined some operations on soft sets in [13]. Aktaş et al. generalized soft sets by defining the concept of soft groups in [2]. After then, QiuMei Sun et al. gave soft modules in [20]. In this paper, the concept of neutrosophic soft module is introduced and some of its basic properties are studied.

Key Words.. Neutrosophic set, neutrosophic soft set, neutrosophic soft modules, neutrosophic soft homomorphism.

Date of Publication: 30.5.2018

DOI: 10.24297/jam.v14i2.7401 ISSN: 2347-1921 Volume: 14 Issue: 02 Journal: Journal of Advances in Mathematics Website: https://cirworld.com This work is licensed under a Creative Commons Attribution 4.0 International License.

1. Introduction The contribution of mathematics to the present-day technology in reaching to a fast trend cannot be ignored. The treories presented differently from classical methods in studies such as fuzzy set [21],intuitionistic fuzzy set [3], soft set [17], neutrosophic set [19], etc. The algebraic structure of set theories dealing with uncertainties has also been studied by some authors.After Molodtsov’s work, some different applications of soft sets were studied in [16].

Maji et al. [14] presented the concept of fuzzy soft set. Rosenfeld [18] proposed the concept of fuzzy groups in order to establish the algebraic structures of fuzzy sets. Aktaş and Çağman [2] defined soft groups and compared soft sets with fuzzy sets and rough sets. After the definition of fuzzy soft group is given by some authors [4,11]. F.Feng et al. [8] gave soft semirings and U.Acar et al. [1] introduced initial concepts of soft rings. Definition of fuzzy module is given by some authors [12,22].

Qiu- Mei Sun et al. [20] defined soft modules and investigated their basic properties. Fuzzy soft modules and intuitionistic fuzzy soft modules was given and researched by C. Gunduz (Aras) and S. Bayramov [9,10].

The main purpose of this paper is to introduce a basic version of neutrosophic soft module theory, which extends the notion of module by including some algebraic structures in soft sets. Finally, we investigate some of neutrosophic soft module basic properties.

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1. Preleminaries In this section, we will give some preliminary information for the present study.

Definition 2.1. [19] A neutrosophic set A on the universe of discourse X is defined as:

A =  x,  (x ), I  (x ), F (x ) : x  X , where



, I , F : X →− 0,1+ and _ 0   (x ) + I  (x ) + F (x )+ 3.

Definition 2.2 [17] Let X be an initial universe, E be a set of all parameters and the power set of X . A pair

F : E → P( X ) .

(F, E ) is called a soft set over

P(X )

denotes

X , where F is a mapping given by

Firstly, neutrosophic soft set defined by Maji [13] and later concept has been modified by Deli and Bromi [7] as given below:

P( X ) ~ denote the set of all neutrosophic sets of X . Then, a neutrosophic soft set F, E over X is a set ~ ~ ~ defined by a set valued function F representing a mapping F : E → P( X ) where F is called ~ approximate function of the neutrosophic soft set F, E . In other words, the neutrosophic soft set is a parameterized family of some elements of the set P( X ) and therefore it can be written as a set Definition 2.3. Let X be an initial universe set and E be as a set of parameters. Let

(

(

)

)

of ordered pairs,

(F~, E ) = (e,

)

x, F~ (e ) (x ), I F~ (e ) (x ), FF~ (e ) (x ), : x  X : e  E



 

where F~ (e ) (x ), I F~ (e ) (x ), FF~ (e ) (x )  0,1 , respectively called the truth-membership, indeterminacymembership, falsity-membership function of F (e ) . Since supremum of each

~

inequality 0  F~ (e ) (x ) + I F~ (e ) (x ) + FF~ (e ) (x )  3 is obvious.

(~ )

, I, F is 1 so the

Definition 2.4. [6] Let F, E be neutrosophic soft set over the common universe

(

)

(

~ ~ complement of F, E is denoted by F, E

(F~, E ) =  (e, c

Obvious that,

(~ )

( X , E ) . The

and is defined by:

)



x, FF~ (e ) (x ),1 − I F~ (e ) (x ), TF~ (e ) (x ), : x  X : e  E .

((F~, E ) ) = (F~, E ) c c

)

c

.

(~ )

Definition 2.5. [13] Let F, E and G, E be two neutrosophic soft sets over the common

~ ~ ( X , E ) . (F, E ) is said to be neutrosophic soft subset of (G, E ) if F~ (e ) (x )  G~ (e ) (x ), I F~ (e ) (x )  I G~ (e ) (x ), FF~ (e ) (x )  FG~ (e ) (x ), e  E, x  X . It is denoted by

universe

(F~, E )  (G~, E ).

7671

The operations of union, intersection, difference, AND, OR on neutrosophic soft sets are defined differently from the studies [6,13]. In addition, basic properties of these operations will be presented.

(~ )

(~ ) ( X , E ) . Then their union is denoted by (F~ , E ) (F~ , E ) = (F~ , E ) and is defined by:

Definition 2.6. Let F1 , E and F2 , E be two neutrosophic soft sets over the common universe 1

(F~ , E ) =  (e, 3

2

3

)

x, F~3 (e ) (x ), I F~3 (e ) (x ), FF~3 (e ) (x ), : x  X : e  E



where

 I ( ) (x ) = max I ( ) (x ), I F ( ) (x ) = min F ( ) (x )F

 ( ) ( x ), ( ) ( x ).

F~3 (e ) (x ) = max F~1 (e ) (x ), F~2 (e ) (x ) , ~ F3 e

~ F1 e

~ F2 e

~ F3 e

~ F1 e

~ F2 e

(~ )

(~ ) ( X , E ) . Then their union is denoted by (F~ , E ) (F~ , E ) = (F~ , E ) and is defined by:

Definition 2.7. Let F1 , E and F2 , E be two neutrosophic soft sets over the common universe 1

(F~ , E ) =  (e, 3

2

3

)

x, F~3 (e ) (x ), I F~3 (e ) (x ), FF~3 (e ) (x ), : x  X : e  E



where

  I ( ) (x ) = min I ( ) (x ), I ( ) (x ), F ( ) (x ) = max F ( ) (x )F ( ) (x ).

F~3 (e ) (x ) = min F~1 (e ) (x ), F~2 (e ) (x ) , ~ F3 e

~ F1 e

~ F3 e

~ F1 e

~ F2 e

~ F2 e

(~ ) (~ ) ( X , E ) . Then ” (F~ , E ) difference (F~ , E )” operation on them is denoted by (F~ , E ) \ (F~ , E ) = (F~ , E ) and is defined by (F~ , E ) = (F~ , E ) (F~ , E ) as follows:

Definition 2.8. Let F1 , E and F2 , E be two neutrosophic soft sets over the common universe 1

2

c

1

2

3

3

(F~ , E ) =  (e, 3

1

2

)

x, F~3 (e ) (x ), I F~3 (e ) (x ), FF~3 (e ) (x ), : x  X : e  E



where

  I ( ) (x ) = min I ( ) (x ),1 − I ( ) (x ), F ( ) (x ) = max F ( ) (x ), T ( ) (x ).

F~3 (e ) (x ) = min F~1 (e ) (x ), FF~2 (e ) (x ) , ~ F3 e

~ F3 e

Definition 2.9. Let

( X , E ) . Then

~ F1 e

~ F1 e

~ F2 e

~ F2 e

(F~ , E ) i  I  be a family of neutrosophic soft over the common universe i

7672

 (F , E ) = (e,

x, sup F~i (e ) (x )

 (F , E ) = (e,

x, inf F~i (e ) (x )

~

i

iI

~

i

iI





, sup I F~i (e ) (x )





, inf I F~i (e ) (x )

(~ )

iI

iI





, inf FF~i (e ) (x )





, sup FF~i (e ) (x )

iI

iI

(~ )

)



)







, : x X :eE ,





, : x X :eE .

iI

iI

Definition 2.10. Let F1 , E and F2 , E be two neutrosophic soft sets over the common universe

( X , E ) . Then ” AND”

(~ ) (~ ) (~

)

operation on them is denoted by F1 , E  F2 , E = F3 , E  E and is

defined by:

(F~ , E  E ) = ((e , e ), 3

1

2

)

x, F~3 (e1 ,e2 ) (x ), I F~3 (e1 ,e2 ) (x ), FF~3 (e1 ,e2 ) (x ), : x  X : (e1 , e2 )  E  E



where

  ) ( x ) = min I ( ) ( x ), I ( ) ( x ), ) ( x ) = max F ( ) ( x )F ( ) ( x ).

F~3 (e1 ,e2 ) (x ) = min F~1 (e1 ) (x ), F~2 (e2 ) (x ) , I F~3 (e1 ,e2

~ F1 e1

FF~3 (e1 ,e2

(~ )

~ F2 e2

~ F1 e1

~ F2 e2

(~ )

Definition 2.11. Let F1 , E and F2 , E be two neutrosophic soft sets over the common universe

( X , E ) . Then ” OR”

(~ ) (~ ) (~

)

operation on them is denoted by F1 , E  F2 , E = F3 , E  E and is defined

by:

(F~ , E  E ) = ((e , e ), 3

1

2

)

x, F~3 (e1 ,e2 ) (x ), I F~3 (e1 ,e2 ) (x ), FF~3 (e1 ,e2 ) (x ), : x  X : (e1 , e2 )  E  E



where

  ) ( x )= max I ( ) ( x ), I ( ) ( x ), ) ( x ) = min F ( ) ( x )F ( ) ( x ).

F~3 (e1 ,e2 ) (x ) = max F~1 (e1 ) (x ), F~2 (e2 ) (x ) , I F~3 (e1 ,e2

~ F1 e1

FF~3 (e1 ,e2 Definition 2.12.

~ F1 e1

(~ )

I F~3 (e ) (x ) = 0,

(~ )

I F~ (e ) (x ) = 1,

( X , E ) is said to be absolute

neutrosophic soft set if

FF~ (e ) (x ) = 0; e  E, x  X . It is denoted by 1( X , E ) .

0 c( X , E ) = 1( X , E ) and 1c( X , E ) = 0 ( X , E ) .

(~ ) (~ )

Proposition 2.1. Let F1 , E , F2 , E and universe

( X , E ) is said to be null neutrosophic

FF~3 (e ) (x ) = 1; e  E, x  X . It is denoted by 0 ( X , E ) .

(2) A neutrosophic soft set F, E over

Clearly

~ F2 e2

(1) A neutrosophic soft set F, E over

soft set if F~ (e ) (x ) = 0, 3

F~ (e ) (x ) = 1,

~ F2 e2

( X , E ) . Then,

(F~ , E ) be two neutrosophic soft sets over the common 3

7673

(1)

(F~ , E ) (F~ , E ) (F~ , E ) = (F~ , E ) (F~ , E ) (F~ , E ) and (F~ , E ) (F~ , E ) (F~ , E ) = (F~ , E ) (F~ , E ) (F~ , E ); (F~ , E ) (F~ , E ) (F~ , E ) = (F~ , E ) (F~ , E ) (F~ , E ) (F~ , E ) and (F~ , E ) (F~ , E ) (F~ , E ) = (F~ , E ) (F~ , E ) (F~ , E ) (F~ , E ); (F~ , E ) 0( ) = (F~ , E ) and (F~ , E ) 0( ) = 0( ) ; (F~ , E )1( ) = 1( ) and (F~ , E )1( ) = (F~ , E ) .

1

2

1

(2)

2

(4)

1

2

3

2

X ,E

3

2

3

2

1

1

X ,E

2

1

3

1

1

1

3

1

1

(3)

3

2

1

X ,E

1

X ,E

1

(~ )

1

X ,E

3

3

X ,E

1

(~ )

Proposition 2.2. Let F1 , E and F2 , E be two neutrosophic soft sets over the common universe

( X , E ) . Then,

(F~ , E ) (F~ , E ) = (F~ , E )  (F~ , E ) ; ~ ~ ~ ~ (2) (F , E ) (F , E ) = (F , E )  (F , E ) . ~ ~ Proposition 2.3. Let (F , E ) and (F , E ) be two neutrosophic soft sets over the common universe c

(1)

1

2

1

2

c

2

c

(1) (2)

c

c

1

1

( X , E ) . Then,

c

1

2

2

(F~ , E ) (F~ , E ) = (F~ , E )  (F~ , E ) ; (F~ , E ) (F~ , E ) = (F~ , E )  (F~ , E ) . c

1

2

1

2

c

c

1

c

2

c

c

1

2

Definition 2.13. Let M be a left R -module and let Then we say

A = (T , I , F ) be a neutrosophic set over M.

(M , T , I , F ) is a neutrosophic modul, if the following conditions are satisfied:

T (0) = I (0) = 1; F (0) = 0 b) T (x + y )  T (x )  T ( y ); I (x + y )  I (x )  I ( y ); F (x + y )  maxF (x ), F ( y ) c) T (x )  T (x ); I (x )  I (x ); F (x )  F (x ) a)

Definition 2.14. Let

(M1 ,T1 , I1 , F1 )

and

(M 2 ,T2 , I 2 , F2 ) be two neutrosophic modules over

M1

and

M 2 , respectively. We say that f is a homomorphism of neutrosophic modules, if the following conditions for homomorphism of f : M 1 → M 2 modules are satisfied: T2 ( f (x ))  T1 (x ), I 2 ( f (x ))  I1 (x ), F2 ( f (x))  F1 (x)

1. Neutrosophic soft modules In this paper R is an ordinary ring. Let M be a left (or right) R -module and let

NS (M ) denotes the family of neutrosophic sets over M .

Definition 3.1. Let

(F,~ A)

be a neutrosophic soft set over M . Then

(F,~ A)

A   be a set.

is said to be a

~ neutrosophic soft module over M iff a  A , F (a ) = (Ta , I a , Fa ) is a neutrosophic submodule ~ of M and denoted as Fa .

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(~ )

(F~ , A)

Definition 3.2. Let

and F , B be two neutrosophic soft modules over M and

1

2

N

respectively, and let f : M → N be a homomorphism of modules, and let g : A → B be a

( f , g ) : (F~1 , A) → (F~ 2 , B)

mapping of sets. Then we say that

is a neutrosophic soft

homomorphism of neutrosophic soft modules, if the following condition is satisfied:

( )

( )

( )

~ ~ ~ f T(1a ) = F 2 (g (a )) = Tg2(a ) , f I (1a ) = F 2 (g (a )) = I g2(a ) , f F(1a ) = F 2 (g (a )) = Fg2(a )

(

) (

~ ~ f : M , F(1a ) → N , Fg2(a )

a  A ,

Note that for

)

is a neutrosophic homomorphism of

neutrosophic modules.

Neutrosophic soft modules and morphisms of their is consists of a category. This category is denoted NSM .

(~ ) (~ ) ~ ~ intersection (F , A)  (F , B ) is a neutrosophic soft module over M .

Theorem 3.1. Let F , A and F , B be two neutrosophic soft modules over M . Then their 1

2

1

2

(F~ , A) (F~ , B) = (F~ , C ) ,where

C = A  B . Since the neutrosophic soft set ~ T = T  T , I = I  I , F = F  F is a neutrosophic submodule, for c  C , F 3 , C is a neutrosophic soft module over M .

Proof. Let 3 c

1 c

1

2 c

2

3 c

1 c

Theorem 3.2. Let

(

) (

3

2 c

3 c

(F~ , A)

and

1

)

1 c

(

2 c

(F~ , B) 2

)

be two neutrosophic soft modules over M . Then

~ ~ F 1 , A  F 2 , B is a neutrosophic soft module over M .

Proof.

(F~ , A)  (F~ , B) = (F~ , A  B).

We can write

submodules

of

1

2

3

~ ~ Fa1  Fb2

M,

is

a

Since

neutrosophic

~ ~ Fa1 and Fb2 are neutrosophic submodule

of

M.

Thus,

T 3 (a, b) = Ta1  Tb2 , I 3 (a, b) = I a1  I b2 , F 3 (a, b) = Fa1  Fb2 is a neutrosophic submodule of M , ~1 ~2 for all (a, b)  A  B . Hence, we find that F , A  F , B is a neutrosophic soft module over M ~2 ~1 .Theorem 3.3. Let F , A and F , B be two neutrosophic soft modules over M . If A  B =  , ~1 ~2 then F , A  F , B is a neutrosophic soft module over M .

(

(

) (

)

) (

)

(F~ , A) (F~ , B) = (F~ , C ) .

Since

(

)

(

)

A  B =  , it follows that either ~ 3 ~1 c  A − B or c  B − A for all c  C . If c  A − B , then Fb = Fb is a neutrosophic submodule ~3 ~2 of M , and if c  B − A , then Fb = Fb is a neutrosophic submodule of M . Hence, ~ ~ F 1 , A  F 2 , B is a neutrosophic soft module over M .

Proof. We can write

(

) (

1

2

3

)

(~ )

(~ )

(~ )

Definition 3.3. Let F , A and F , B be two neutrosophic soft modules over M . Then F , A 1

1

2

(~ )

is called a neutrosophic soft submodule of F , B if 2

1) A  B 2) For all a  A ,

(

~ Fa1 = Ta1 , I a1 , Fa1

)

is a neutrosophic submodule of

Ta1  Ta2 , I a1  I a2 , Fa1  Fa2 .

7675

(

~ Fa2 = Ta2 , I a2 , Fa2

)

,

(~ ) (~ ) ~ ~ all a  A , then (F , A) is a neutrosophic soft submodule of (F , A) .

Theorem 3.4. Let F , A and F , A be two neutrosophic soft modules over M . If 1

2

1

~ ~ Fa1  Fa2 for

2

Proof. The proof of the theorem is straightforward.

(~ )

Theorem 3.5. Let F, A be a neutrosophic soft modules over M , and let

(

)

(F~ , A ) i

~ nonempty family of neutrosophic soft submodules of F, A . Then 1)

i

iI

be

 (F , A ) is a neutrosophic soft submodule of (F, A), ~

~

i

i

iI

2)

(

)

(

)

~ ~  Fi , Ai is a neutrosophic soft submodule of F, A ,

iI

3) If Ai  A j =  , for all i, j  I , then

(

)

(

)

~ ~  Fi , Ai is a neutrosophic soft submodule of F, A .

iI

( ~ ) and (F~ , B) be two neutrosophic soft modules over M and N respectively, and ( f , g ) : (F~ , A) → (F~ , B) be a neutrosophic soft homomorphism of these modules.

Let F , A 1

2

1

2

Now in this section, we introduce the kernel and image of neutrosophic soft homomorphism of

~ Define by F  : A → NS (M ) M  = ker f . ~ Ta = Ta M  , I a = I a M  , Fa = Fa M  . Then F , A is a neutrosophic soft module over M  . It is ~ clear that this module is a neutrosophic soft submodule of F, A . neutrosophic

soft

modules.

Let

(~ )

Definition 3.4. F , A is said to be kernel of

(

)

(

)

( f , g ) and denoted by ker ( f , g ) .

B = g ( A) . Then for all b  B , there exists a  A such that g (a ) = b . Let ~ We define the mapping as F  2 : B → NS (N ) N  = Im f  N .

2 2 2 2 2 2 T  (b) = T (g (a )) N  , I  (b) = I (g (a )) N  , F  (b) = F (g (a )) N  . Since ( f , g ) is a

Now, let

( )

( )

( )

f Ta1 = Tg2(a ) , f I a1 = I g2(a ) , f Fa1 = Fg2(a ) is satisfied for all ~ ~ a  A . Then the pair F  2 , B is a neutrosophic soft module over N  and F  2 , B is a ~2 neutrosophic soft submodule of F , B . neutrosophic soft homomorphism,

(

(~

)

) (

)

Definition 3.5. F  , B  is said to be image of 2

(

)

( f , g ) and denoted by Im( f , g ) .

(~ )

Proposition 3.1. Let F, A . be a neutrosophic soft module over M , N be an R − module and

(( ) )

~ f : M → N . be a homomorphism of R − modules. Then f F , A . is a neutrosophic soft module over N .

( ~)

Proof. If the mapping f F : A → NS (N ) is defined by

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( f (T ))a ( y ) = supTa (x ) : f (x ) = y ( f (I ))a ( y ) = supI a (x ) : f (x ) = y , ( f (F ))a ( y ) = inf Fa (x ) : f (x ) = y the proof is completed.

( f ,1A ) : (F~, A) → ( f (F~ ), A) is a neutrosophic soft homomorphism of neutrosophic soft

Note that modules.

(F,~ A) is a neutrosophic soft module over N and ~ is a homomorphism of R − modules, then ( f (F ), A) is a neutrosophic soft module

Proposition 3.2. If M is an R − module,

f :M → N over M .

−1

Proof. If the mapping f

−1

(F~): A → NS (M ) is defined by

( f (T )) (x) = T ( f (x)), ( f (I )) (x) = I ( f (x)), ( f (F )) (x) = F ( f (x)) , −1

−1

−1

a

a

a

a

a

a

the proof is completed.

It is clear that

( f ,1A ) : ( f −1 (F~ ), A) → (F~, A) is a neutrosophic soft homomorphism of neutrosophic

soft modules.

Lemma 3.1. Let M and N be an R − modules and f : M → N be an R − homomorphism and

(F~ , A) and (F~ , A) are two neutrosophic soft modules over M 1

(i)

2

and N respectively.

( f ,1A ) : (F~1 , A) → (F~ 2 , A) is a neutrosophic soft homomorphism if and only if

( )

( )

( ) is satisfied.

for all a  A ,

T  f T ,I  f I ,F  f F 2 a

(ii)

1 a

2 a

1 a

2 a

1 a

( f ,1A ) : (F~1 , A) → (F~ 2 , A) is a neutrosophic soft homomorphism if and only if

T  f 1 a

−1

(T ), I 2 a

1 a

Theorem 3.6. If

( iI

)

−1

 f

(I ), F 2 a

1 a

(F~ , A ) i

i

iI

 f

−1

(F ) is satisfied .

for all a  A ,

2 a

is a family of neutrosophic soft modules over

~ Fi , Ai is a neutrosophic soft module over  M i . iI

~

Proof. Define F :

A

i

iI

~ →  M i F = (T , I , F ) by iI

T (ai ) =  pi−1 (Ti )ai , I (ai ) =  pi−1 (I i )ai , F (ai ) =  pi−1 (Fi )ai , iI

where

iI

iI

pi :  M i → M i is a projection mapping. Since iI

p

−1 i

(Ti )a :  M i → 0,1, pi−1 (I i )a :  M i → 0,1, pi−1 (Fi )a :  M i → 0,1 i

iI

i

i

iI

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iI

M i iI ,

then

 pi−1 (Ti )ai ,  pi−1 (I i )ai ,  pi−1 (Fi )ai is

 M ,for all i  I , also a neutrosophic soft module over  M . is a neutrosophic soft module over

i

iI

iI

iI

iI

i

iI

Theorem 3.7. If

(F~ , A ) i

i

is a family of neutrosophic soft modules over the family of modules

iI

M i iI , then i (F~i , Ai ) is a neutrosophic soft module over iI M i . I Proof.

~ F = (T , I , F )

~ F :  Ai →  M i

Define

iI

iI

for

T (ai ) =  ji (Ti )ai , I (ai ) =  ji (I i )ai , F (ai ) =  ji (Fi )ai iI

iI

iI

ai   Ai

all

by

iI

ji : M i →  M i is a

where

iI

embedding mapping. Since ji (Ti )a , ji (I i )a , ji (Fi )a is a neutrosophic soft submodule over  M i i

for all

i

iI

i

i  I , F (ai ) is a neutrosophic submodule over  M i . iI

M i iI

Lemma 3.2. 1) Given modules

(

~ A =  f i : M i → N iI . If Fi , Ai a neutrosophic soft module

)

iI

and

and a family of

N

are neutrosophic soft modules over

R − homorphisms

M i iI , then there exist

 ~2   F ,  Ai  over N such that for all i  I , iI  

(

)

~  ~ f i : Fi , Ai →  F 2 ,  Ai  iI   is a neutrosophic soft homomorphism of neutrosophic soft modules.

M and N i iI and a family of R − homorphisms B = gi : M → Ni iI . If

2) Given modules

(

)

module

~   F,  Ai  over M such that for all i  I ,  iI 

~ Fi 2 , Bi

iI

Ni iI ,

are neutrosophic soft modules over

(

then there exist a neutrosophic soft

~  ~ g i :  F ,  Ai  → Fi 2 , Bi  iI 

)

is a neutrosophic soft homomorphism of neutrosophic soft modules.

Proof. 1) Define

T

2

~ ~ F 2 :  Ai → N , F 2 = (T 2 , I 2 , F 2 ) by

(ai ) = i f i (Ti

2) Define

2

iI

)

ai

( )

, I 2 (ai ) =  f i I i2 i

ai

( )

, F 2 (ai ) =  f i Fi 2 i

ai

.

~ ~ F :  Ai → M , F = (T , I , F ) by iI −1 i

T (ai ) =  g iI

(Ti )a , I (ai ) = iI g i−1 (I i )a , F (ai ) = iI g i−1 (Fi )a i

i

i

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for all

ai   Ai . iI

By using this lemma, we define the concepts of submodule , quotient module, product and coproduct operations in the category of neutrosophic soft modules.

(~ )

Corallary 3.1. If F, A is a neutrosophic soft module over M and N is a submodule of M and

( ( ) )

~ i : N → M is a embedding mapping, then i −1 F , A is a neutrosophic soft module over N .

(~ ) ~ projection, then ( p(F ), A) is a neutrosophic soft module over quotient module M

Corallary 3.2. If F, A is a neutrosophic soft module over M and p : M → M ~ is a canonical

If

(F~ , A ) i

i

iI

M i iI , then we

is a family of neutrosophic soft modules over the family of modules

can define the product and coproduct of these families by

 (F , A ) ~

i

i

and

iI

~.

(

)

~  Fi , Ai , respectively.

iI

Theorem 3.8. The category of neutrosophic soft modules has zero objects, sums, product, kernel and cokernel.

(~ )

(~ )

Let M and N be ,respectively, right and left modules over R (ring). Let F , A and F , B be 1

2

two neutrosophic soft modules over M and N , respectively. We consider tensor product of modules as M  N . The mapping

~ ~ F1  F 2 : A  B → M  N is defined by

(T  T )(a, b) = T (a )  T (b), (I  I )(a, b) = I (a )  I (b), (F  F )(a, b) = F (a )  F (b) 1

2

1

1

2

1

2

1

2

2

1

2

(a, b)  A  B .

for 

(~

(~ )

)

~2

(~ )

Definition 3.6. F  F , A  B is said to be tensor product of F , A and F , B and denoted 1

(~ ) (~ )

1

2

by F , A  F , B .

1

2

(~

)

~2

Theorem 3.9. F  F , A  B is a neutrosophic soft module over M  N .

1

(

)

(~ ~ ~ ~ denoted by (F , A)  (F , B ).

)

(

)

~ ~ ~ ~ (a, b)  A  B , M , Fa1 and N , Fb2 are neutrosophic soft modules. Fa1  Fb2 is a ~1 ~ 2 neutrosophic submodule over M  N and F  F , A  B is a neutrosophic soft module over M N. Proof. For

(

)

(~ )

(~ )

Definition 3.7. F  F , A  B is said to be tensor product of F , A and F , B , and 1

1

2

1

2

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2

2. Conclusion This paper summarized the basic concepts of neutrosophic soft sets and neutrosophic soft modules.

By using these concepts, we studied the algebraic properties of neutrosophic soft sets in module structure.

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