RESIDUAL QUOTIENT AND ANNIHILATOR OF INTUITIONISTIC FUZZY SETS OF RING AND MODULE

International Journal of Computer Science & Information Technology (IJCSIT) Vol 9, No 4, August 2017

RESIDUAL QUOTIENT AND ANNIHILATOR OF INTUITIONISTIC FUZZY SETS OF RING AND MODULE P.K. Sharma1 and Gagandeep Kaur2 1

P.G. Department of Mathematics, D.A.V. College, Jalandhar City, Punjab, India 2 Research Scholar, IKG PT University, Jalandhar , Punjab

ABSTRACT In this paper, we introduce the concept of residual quotient of intuitionistic fuzzy subsets of ring and module and then define the notion of residual quotient intuitionistic fuzzy submodules , residual quotient intuitionistic fuzzy ideals. We study many properties of residual quotient relating to union, intersection, sum of intuitionistic fuzzy submodules (ideals). Using the concept of residual quotient, we investigate some important characterization of intuitionistic fuzzy annihilator of subsets of ring and module. We also study intuitionistic fuzzy prime submodules with the help of intuitionistic fuzzy annihilators. Many related properties are defined and discussed.

KEYWORDS Intuitionistic fuzzy (prime) submodule (ideal), residual quotient intuitionistic fuzzy submodules (ideal), intuitionistic fuzzy annihilator, semiprime ring.

1. INTRODUCTION The concept of intuitionistic fuzzy sets was introduced by Atanassov [1], [2] as a generalization to the notion of fuzzy sets given by Zedah [16]. Biswas was the first to introduce the intuitionistic fuzzification of the algebraic structure and developed the concept of intuitionistic fuzzy subgroup of a group in [5]. Hur and others in [8] defined and studied intuitionistic fuzzy subrings and ideals of a ring. In [7] Davvaz et al. introduced the notion of intuitionistic fuzzy submodules which was further studied by many mathematicians (see [4], [9], [12], [13], [14]).

The correspondence between certain ideals and submodules arising from annihilation plays a vital role in the decomposition theory and Goldie like structures (see [6]). A detailed study of the fuzzification of this and related concepts can be found in [10], [11] and [15]. Intuitionistic fuzzification of such crisp sets leads us to structures that can be termed as intuitionistic fuzzy prime submodules. In this paper, we attempt to define annihilator of an intuitionistic fuzzy subset of a module using the concept of residual quotients and investigate various characteristic of it.

This concept will help us to explore and investigate various facts about the intuitionistic fuzzy aspects of associated primes, Godlie like structures and singular ideals.

2. PRELIMINARIES Throughout this section, R is a commutative ring with unity 1, 1 ≠ 0, M is a unitary R-module and θ is the zero element of M. The class of intuitionistic fuzzy subsets of X is denoted by IFS(X).

Definition (2.1)[4] Let R be a ring. Then A ∈ IFS(R) is called an intuitionistic fuzzy ideal of R if for all x , y ∈ R it satisfies DOI:10.5121/ijcsit.2017.9401

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International Journal of Computer Science & Information Technology (IJCSIT) Vol 9, No 4, August 2017

(i ) µ A ( x − y ) ≥ µ A ( x) ∧ µ A ( y ) , ν A ( x − y ) ≤ ν A ( x) ∨ ν A ( y ); (ii ) µ A ( xy ) ≥ µ A ( x) ∨ µ A ( y )

, ν A ( xy ) ≤ ν A ( x) ∧ν A ( y ).

The class of intuitionistic fuzzy ideals of R is denoted by IFI(R).

Definition (2.2)[4] An intuitionistic fuzzy set A of an R-module M is called an intuitionistic fuzzy submodule (IFSM) if for all x , y ∈ M and r∈ R , we have

(i ) µ A (θ ) = 1 , ν A (θ ) = 0; (ii ) µ A ( x + y ) ≥ µ A ( x) ∧ µ A ( y ) , ν A ( x + y ) ≤ ν A ( x) ∨ ν A ( y ); (iii ) µ A (rx) ≥ µ A ( x) , ν A (rx) ≤ ν A ( x).

The class of intuitionistic fuzzy submodules of M is denoted by IFM(M).

Definition (2.3)[2, 12] Let α, β ∈ [0, 1] with α + β ≤ 1. An intuitionistic fuzzy point, written as x(α,β), is defined to be an intuitionistic fuzzy subset of X, given by

(α , β ) ; if y = x x(α , β ) ( y ) =  . We write x(α , β ) ∈ A if and only if x ∈ C(α , β ) ( A),  (0,1) ; if y ≠ x where C(α , β ) ( A) = { x ∈ X : µ A (x) ≥ α and ν A (x) ≤ β } is the (α , β ) − cut set (crisp set ) of the intuitionistic fuzzy set A in X. Definition (2.4)[9] Let M be an R-module and let A, B ∈ IFM(M). Then the sum A + B of A and B is defined as

µ A (y) ∧ µ B (z) ; if x = y + z 0 ; otherwise 

µ A+ B ( x) = 

ν (y) ∨ν B (z) ; if x = y + z and ν A+ B ( x) =  A 1 ; otherwise 

Then, A + B∈IFM(M).

Definition (2.5) Let M be an R-module and let A∈ IFS(R) and B∈IFM(M). Then the product AB of A and B is defined as

µ (r) ∧ µB (m) µAB ( x) =  A 

0

; if x = rm ν (r) ∨ν B (m) ; if x = rm and ν AB ( x) =  A , r ∈ R, m ∈ M . ; otherwise 1 ; otherwise 

Clearly, AB∈IFM(M).

Definition (2.6) [9, 12] Let M be an R-module and let A, B ∈IFM(M). Then the product AB of A and B is defined as

µ ( y) ∧ µB (z) µAB (x) =  A 

0

; if x = yz ν ( y) ∨ν B (z) ; if x = yz and ν AB (x) =  A ,where y, z ∈M. ; otherwise 1 ; otherwise  2

International Journal of Computer Science & Information Technology (IJCSIT) Vol 9, No 4, August 2017

Definition (2.7) [3] An P∈IFI(R) is called an intuitionistic fuzzy prime ideal of R if for any A, B ∈ IFI(R) the condition AB ⊆ P implies that either A ⊆ P or B ⊆ P.

Definition (2.8)

(

Let X be a non empty set and A ⊂ X. Then an intuitionistic fuzzy set χ A = µ χ A ,ν χ A

) is called

an intuitionistic fuzzy characteristic function and is defined as

1 ; if x ∈ A 0 ; if x ∉ A

µχ ( x) =  A

0 ; if x ∈ A and ν χ A ( x) =  . 1 ; if x ∉ A

Definition (2.9)

(

)

Let M be R-module and χθ is an IFS on M defined as χθ (x) = µχθ (x),ν χθ (x) ,

1 ; if x = θ where µχθ ( x) =  0 ; if x ≠ θ

0 ; if x = θ and ν χθ ( x) =  1 ; if x ≠ θ

and χ0 and χR are IFSs on R defined by

χ 0 (r ) = ( µ χ (r ),ν χ (r ) ) and χ R (r ) = ( µ χ (r ),ν χ (r ) ) ,where 0

0

R

R

 1 ; if r = 0  0 ; if r = 0 ; ν χ0 (r ) =  and µ χ R ( r ) = 1 ; ν χ R ( r ) = 0, ∀ r ∈ R.  0 ; if r ≠ 0  1 ; if r ≠ 0

µ χ (r ) =  0

Theorem (2.10) Let x ∈ R and α , β ∈ (0,1] with α + β ≤ 1. Then < x(α,β) > = (α,β)< x >,

(α , β ) ; if y ∈< x > where (α , β )< x > ( y ) =  is called the (α , β ) − level (or cut set ) intuitionistic  (0,1) ; if y ∉< x > fuzzy ideal corresponding to < x >.

Proof. Case(i) When y ∈< x > and let y = xn , for some positive interget n, then

µ(α ,β )

(y) = α = µx(α ,β ) (x) ≤ µx(α ,β ) (xn ) = µx(α ,β ) ( y) and

ν (α ,β )

(y) = β = ν x(α ,β ) (x) ≥ ν x(α ,β ) (xn ) = ν x(α ,β ) ( y)

<x >

<x >

Case(ii) When y ∉< x >, then

µ(α ,β )

<x >

(y) = 0 = µx(α ,β ) ( y) and ν (α ,β )< x > (y) = 1 = ν x(α ,β ) ( y).

Thus in both the cases we find that (α , β )< x > ⊆ x(α ,β ) .

Now < x(α ,β ) > = ∩{ A : A ∈ IFI (R) such that x(α ,β ) ⊆ A} implies that (α , β )< x > = < x(α ,β ) > .

3. Residual quotient of intuitionistic fuzzy sets of ring and module Throughout the paper, R is a commutative ring with unity 1, 1 ≠ 0, M is a unitary R-module and θ is the zero element of M. For a subset X of M, recall that < X > denotes the submodules of M generated by X. For any x∈ X, we let < x > denote the submodule of M generated by {x}.

For any submodule N of M we have (N : M) = { r | r∈R, rM ⊆ N }.

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International Journal of Computer Science & Information Technology (IJCSIT) Vol 9, No 4, August 2017

In this section, we introduce and study the notion of residual quotient of intuitionistic fuzzy subsets of ring and module.

Definition (3.1) For A, B ∈ IFS(M) and C∈IFS(R), define the residual quotient (A: B) and (A: C) as follows:

U {D : ( A : C ) = U {E :

D ∈ IF S ( R ) su ch th at D ⋅ B ⊆ A } an d

( A : B) =

E ∈ IF S ( M ) su ch th at C ⋅ E ⊆ A }

Clearly, (A: B) ∈ IFS(R) and (A: C) ∈ IFS(M).

Theorem (3.2) For A, B ∈ IFS(M) and C∈IFS(R). Then we have

(i) (A : B) = U{r(α , β ) : r ∈ R, α , β ∈[0,1] with α + β ≤ 1 such that r(α , β ) ⋅ B ⊆ A} and (ii) (A : C ) = U{ x(α , β ) : x ∈ M , α , β ∈[0,1] with α + β ≤ 1 such that C ⋅ x(α , β ) ⊆ A} .

P ro o f. (i) W e k n o w th at

{r

: r ∈ R , α , β ∈ [0,1] w ith α + β ≤ 1 su ch th at r( α , β ) ⋅ B ⊆ A } ∈ IF S (R )

(α , β )

{r

∴

: r ∈ R , α , β ∈ [0,1] w ith α + β ≤ 1 su ch th at r( α , β ) ⋅ B ⊆ A }

(α , β )

⊆ {D : D ∈ IF S ( R ) su ch th at D ⋅ B ⊆ A } ⇒

U {r α β (

,

)

: r ∈ R , α , β ∈ [0,1] w ith α + β ≤ 1 su ch th at r( α , β ) ⋅ B ⊆ A } ⊆

U {D :

D ∈ IF S ( R ) su ch th at D ⋅ B ⊆ A } = ( A : B ).

Let D ∈ IFS(R) such that D ⋅ B ⊆ A. Let r ∈ R and D(r ) = (α , β ) , i.e., µ D ( r ) = α and ν D ( r ) = β .

Now ,

(r

(α , β )

(

)

B ) ( x ) = µ r( α ,β ) B ( x ), ν r( α ,β ) B ( x ) , where

{

}

µ r α β B ( x ) = ∨ µ r α β ( s ) ∧ µ B ( y ) : s ∈ R , y ∈ M , sy = x (

, )

(

, )

≤ ∨ {µ D ( r ) ∧ µ B ( y ) : y ∈ M , ry = x} [ Q µ r( α , β ) ( s ) ≤ µ D ( r ) = α ] = ∨ {µ D ( s ) ∧ µ B ( y ) : s ∈ R , y ∈ M , sy = x} = µ DB ( x ) ≤ µ A ( x) i.e., µ r( α ,β ) B ( x ) ≤ µ A ( x ), ∀ x ∈ M .

Similarly, we can show that ν r( α ,β ) B ( x ) ≥ ν A ( x ), ∀ x ∈ M . Thus, r(α , β ) B ⊆ A. So, (A:B) ⊆ U {r(α , β ) : r ∈ R , α , β ∈ [0,1] with α + β ≤ 1 such that r(α , β ) B ⊆ A} Hence (A:B) =

U {r α β

( , )

: r ∈ R , α , β ∈ [0,1] with α + β ≤ 1 such that r(α , β ) B ⊆ A}.

(ii) The proof is similar to (i).

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International Journal of Computer Science & Information Technology (IJCSIT) Vol 9, No 4, August 2017

Theorem (3.3) For A, B ∈ IFS(M) and C∈IFS(R). Then we have (i)

(A : B) ⋅ B ⊆ A;

(ii) C ⋅ (A : C) ⊆ A; (iii) C ⋅ B ⊆ A ⇔ C ⊆ (A : B) ⇔ B ⊆ (A : C).

Proof. (i) Now,

( (A : B) ⋅ B) ( x) = ( µ(A:B)⋅B ( x), ν (A:B)⋅B ( x) ) ,

where

µ(A:B)⋅B ( x) = ∨ {µ(A:B) (r ) ∧ µ B ( y ) | r ∈ R, y ∈ M , ry = x} = ∨ {∨ {µ D (r ) | D ∈ IFS (R), D⋅ B ⊆ A} ∧ µ B ( y ) | r ∈ R, y ∈ M , ry = x} = ∨ {µ D (r ) ∧ µ B ( y ) | D ∈ IFS (R), D⋅ B ⊆ A, y ∈ M , ry = x} ≤ ∨ {µ D⋅B (ry ) | D ∈ IFS (R), D⋅ B ⊆ A, y ∈ M , ry = x} ≤ ∨ {µ A (ry ) | y ∈ M , ry = x} = µ A ( x).

Thus, µ(A:B)⋅B ( x) ≤ µ A ( x), ∀ x ∈ M .

Similarly, we can show that ν (A:B)⋅B ( x) ≥ ν A ( x), ∀ x ∈ M .

Hence (A : B) ⋅ B ⊆ A. (ii) The proof is similar to (i) (iii) This is an immediate consequences of (i) and (ii).

Theorem (3.4) For Ai (i ∈ J) , B∈IFS(M) and C∈IFS(R). Then we have     (i)  I Ai  : B = I ( Ai : B) (ii)  I Ai  : C = I ( Ai : C ). i∈J i∈J  i∈J   i∈J      Proof .(i)  I Ai  : B = U D : D ∈ IFS ( R) such that D ⋅ B ⊆ I Ai  i∈J  i∈J    = U {D : D ∈ IFS ( R) such that D ⋅ B ⊆ Ai , ∀i ∈ J }

= U {D : D ∈ IFS ( R) such that D ⊆ ( Ai : B) , ∀i ∈ J }   = U D : D ∈ IFS ( R ) such that D ⊆ I ( Ai : B)   i∈J  ⊆ I ( Ai : B). i∈J

  Thus  I Ai  : B ⊆ I ( Ai : B). i∈J  i∈J  By previous theorem, we have      I (A i : B) ⋅ B  ( x) =  µI (Ai :B)⋅B ( x), ν I (Ai :B)⋅B ( x)  , where  i∈J   i∈J i∈J  5

International Journal of Computer Science & Information Technology (IJCSIT) Vol 9, No 4, August 2017

µ

I (A i :B)⋅B i∈J

   ( x ) = ∨  I µ (A i :B) ( r )  ∧ µ B ( y ) | r ∈ R , y ∈ M , ry = x    i∈J 

) { ( ≤ ∨{ ∧ (µ ( ry ) ) | r ∈ R , y ∈ M , ry = x} ≤ ∨ { ∧ µ ( ry ) | r ∈ R, y ∈ M , ry = x}

}

≤ ∨ ∧ µ (A i :B) ( r ) ∧ µ B ( y ) | r ∈ R, y ∈ M , ry = x i∈J

(A i :B)

i∈J

i∈J

=µ

I Ai

Ai

( x)

i∈J

i.e., µ

I (Ai :B)⋅B

( x) ≤ µ

i∈J

I Ai

( x ), ∀ x ∈ M .

i∈J

Similarly, we can show that ν

I (A i :B)⋅B

( x) ≥ ν

i∈J

I Ai

( x ), ∀ x ∈ M .

i∈J

  : B) ⊆  I A i  : B .......(2) i∈J i∈J i∈J  i∈J    From (1) and (2) we get I (A i : B) =  I A i  : B. i∈J  i∈J 

Thus ,

I (A

i

: B) ⋅ B ⊆ I A i and so

I (A

i

(ii) The proof is similar to (i).

Theorem (3.5) For A, B ∈IFS(M) and C∈ IFS(R) ( i ) If A ∈ IF M ( M ), th en ( ii ) If C ∈ IF I ( R ), th en

Proof. (i) Clearly,

( A : B ) = U {D : D ∈ IF I ( R ) su ch th at D ⋅ B ⊆ A } ( A : C ) = U {E : E ∈ IF M (M ) su ch th at C ⋅ E ⊆ A }

U{D: D∈IFI (R) s.t D⋅ B ⊆ A} ⊆ U{D: D∈IFS(R) s.t D⋅ B ⊆ A} = (A: B).

Let r ∈ R, α , β ∈ (0,1] with α + β ≤ 1 such that r(α , β ) ⋅ B ⊆ A. Let D = < r(α , β ) > . Then < r(α , β ) > ⋅B = (α , β )< r > ⋅ B.

{

Again, µ (α , β )< r > B ( x ) = ∨ µ (α , β )< r > ( s ) ∧ µ B ( y ) | r ∈ R , y ∈ M , sy = x

}

= ∨ {α ∧ µ A ( y ) | s ∈< r > , y ∈ M , sy = x}

{

≤ ∨ µ r( α ,β ) B ( ry ) | t ∈ R , y ∈ M , t(ry ) = x

}

≤ ∨ {µ A ( ry ) | t ∈ R , y ∈ M , t(ry ) = x} ≤ ∨ {µ A (t( ry )) | t ∈ R , y ∈ M , t(ry ) = x} = µ A ( x ).

Thus, µ(α , β )<r> B ( x) ≤ µ A ( x).Similarly, we can show that ν (α , β )<r > B ( x) ≥ ν A ( x), ∀x ∈ M .

Therefore, we have (α , β )< r > ⋅ B ⊆ A. Hence Hence

U{D : D ∈ IFI ( R) such that D ⋅ B ⊆ A} ⊇ ( A : B) ( A : B) = U {D : D ∈ IFI ( R) such that D ⋅ B ⊆ A}.

( ii ) The proof is similar to ( i ) .

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International Journal of Computer Science & Information Technology (IJCSIT) Vol 9, No 4, August 2017

Theorem (3.6) For A∈IFM(M), B∈IFS(M) and C∈IFI(R). Then (i) (A:B)∈IFI(R) and (ii) (A:C)∈IFM(M).

Proof. (i) Since χ0A ⊆ χθ ⊆ A, so χ0 ⊆ (A: B). Let r1 , r2 , r, s ∈ R be any elements. Then,

µ( A:B ) (r1 ) ∧ µ( A:B ) (r2 )

( {

}) ( {

})

= ∨ µ A1 (r1 ) : A1 ∈ IFI ( R), A1 A ⊆ B ∧ ∨ µ A2 (r2 ) : A2 ∈ IFI ( R), A2 A ⊆ B

{

}

= ∨ µ A1 (r1 ) ∧ µ A2 (r2 ) : A1 , A2 ∈ IFI ( R), A1 A ⊆ B, A2 A ⊆ B

{ ≤ ∨ {µ(

}

≤ ∨ µ( A1 + A2 ) (r1 ) ∧ µ( A1 + A2 ) (r2 ) : A1 , A2 ∈ IFI ( R), A1 A ⊆ B, A2 A ⊆ B A1 + A2 )

}

(r1 − r2 ) : A1 + A2 ∈ IFI ( R), A1 A + A2 A ⊆ ( A1 + A2 ) A ⊆ B + B = B

≤ ∨ {µ D (r1 − r2 ) : D ∈ IFI ( R), DA ⊆ B} = µ( A:B ) (r1 − r2 ).

Thus, µ( A:B ) (r1 − r2 ) ≥ µ( A:B ) (r1 ) ∧ µ( A:B ) (r2 ).

Similarly, we can show that ν ( A:B ) (r1 − r2 ) ≤ ν ( A:B ) (r1 ) ∧ν ( A:B ) (r2 ).

Again, µ( A:B) (sr) = ∨{µD (sr): D ∈ IFI (R), DA ⊆ B} ≥ ∨{µD (r): D ∈ IFI (R), DA ⊆ B} = µ( A:B) (r).

Thus, µ( A:B) (sr) ≥ µ( A:B) (r). Similarly, we can show that ν ( A:B) (sr) ≤ν ( A:B) (r), ∀ r, s ∈ R. Hence ( A : B) ∈ IFI (R). (ii) The proof is similar to (i).

Let A, B∈IFM(M) and C∈IFI(R). Then (A:C) is called the residual quotient intuitionistic fuzzy submodule of A and C and (A:B) is called the residual quotient intuitionistic fuzzy ideal of A and B respectively.

Theorem (3.7) For A , Bi ∈IFS(M) and Ci∈IFS(R) for i ∈ J. Then we have     (i) A:  U Bi  = I ( A : Bi ) (ii) A:  U Ci  = I ( A : Ci )  i∈J  i∈J  i∈J  i∈J Proof .(i) By definition (3.1)       A :  U Bi  = U D : D ∈ IFS ( R ) such that D ⋅  U Bi  ⊆ A  i∈J   i∈J      = U  D : D ∈ IFS ( R ) such that U ( D : Bi ) ⊆ A i∈J   ⊆ U {D : D ∈ IFS ( R ) such that ( D : Bi ) ⊆ A} , ∀i ∈ J = ( A: Bi ), ∀i ∈ J .   Therefore, A :  U Bi  ⊆ I ( A: Bi ).  i∈J  i∈J 7

International Journal of Computer Science & Information Technology (IJCSIT) Vol 9, No 4, August 2017

By T heorem (3.3)      I ( A: B i )  ⋅  U B i  =  i∈ J   i∈ J  T hus

I ( A: B ) ⊆ i

i∈ J





U   I ( A: B )  ⋅ B j∈ J



i

i∈ J

j

 ⊆ 

U ( ( A: B

j

)⋅Bj )⊆

j∈ J

    A :  U B i  . Hence A :  U B i  =  i∈ J   i∈ J 

U A = A. j∈ J

I ( A: B ). i

i∈ J

Lemma (3.8) Let M be a R-module, then ( χθ : χθ ) = χ R .

Proof . Since ( χθ : χθ ) = U {D : D ∈ IFS ( R ) such that D ⋅ χθ ⊆ χθ } ⊆ χ R .

We claim that χ R ⊆ ( χθ : χθ ). For this, we first show that χ R χθ = χθ .

Now, µ χ R χθ ( x) = ∨ { χ R (r ) ∧ χθ (m) : r ∈ R, m ∈ M , rm = x} and

νχ

R χθ

( x) = ∧ { χ R (r ) ∨ χθ (m) : r ∈ R, m ∈ M , rm = x} .

Now, µ χ R χθ ( x) = ∨ { χ R (r ) ∧ χθ (m) : r ∈ R, m ∈ M , rm = x} = ∨ {χθ (m) : r ∈ R, m ∈ M , rm = x} [ Q if x = θ ⇒ one m = θ ] 1 ; if x = θ = [ Q if x ≠ θ ⇒ m ≠ θ ] 0 ; if x ≠ θ Also, ν χ R χθ ( x) = ∧ { χ R (r ) ∨ χθ (m) : r ∈ R, m ∈ M , rm = x} = ∧ {χθ (m) : r ∈ R, m ∈ M , rm = x} 0 ; if x = θ = . 1 ; if x ≠ θ Thus, χ R χθ ( x) = χθ ( x).

So, χ R ⊆ U {D : D ∈ IFS ( R), Dχθ ⊆ χθ } = ( χθ : χθ ).

Hence ( χθ : χθ ) = χ R .

Lemma (3.9) Let M be a R-module, then χ 0 ⊆ ( χθ : A), ∀ A ∈ IFS ( M ).

P ro of. W e sh ow that χ 0 A ⊆ χ θ ∀ A ∈ IF S ( M ).

N ow , µ χ0 A ( x ) = ∨

{µ

χ0

( r ) ∧ µ A ( m ) : r ∈ R , m ∈ M , rm = x

}

W hen x ≠ θ ⇒ r ≠ 0 ; ∀ r ∈ R , such that rm = x ⇒ µ χ 0 ( r ) = 0 ∀ r ∈ R , such that rm = x . S o, µ χ 0 A ( x ) = 0 = µ χ θ ( x ).

W hen x = θ ⇒ µ χ 0 A (θ ) ≤ 1 = µ χ θ (θ ). T hu s, µ χ 0 A ( x ) ≤ µ χ θ ( x ).

S im ilarly, w e can sho w th at ν χ 0 A ( x ) ≥ ν χ θ ( x ). T herefore, χ 0 A ⊆ χ θ .

H en ce χ 0 ⊆

U {D :

D ∈ IF S ( R ) such th at D .A ⊆ χ θ } = ( χ θ : A ).

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International Journal of Computer Science & Information Technology (IJCSIT) Vol 9, No 4, August 2017

Lemma (3.10) Let M be a R-module and A, B∈IFS(M). If A ⊆ B, then ( χ θ : B ) ⊆ ( χ θ : A).

Proof. Let A, B ∈ IFS (M), D ∈ IFS (R). Then DA( x ) = ( µ DA ( x ), ν DA ( x ) ) , where

µ DA ( x ) = ∨ {µ D ( r ) ∧ µ A ( m ) : r ∈ R , m ∈ M , rm = x} and ν DA ( x ) = ∧ {ν D ( r ) ∨ ν A ( m ) : r ∈ R , m ∈ M , rm = x} .

Now, µ D ( r ) ∧ µ A ( m ) ≤ µ D ( r ) ∧ µ B ( m ).

Therefore, µ DA ( x ) = ∨ {µ D ( r ) ∧ µ A ( m ) : r ∈ R , m ∈ M , rm = x} ≤ ∨ {µ D ( r ) ∧ µ B ( m ) : r ∈ R , m ∈ M , rm = x} = µ DB ( x ).

Similarly, we can show that ν DA ( x) ≥ ν DB ( x). Thus DA ⊆ DB. So, DB ⊆ χθ ⇒ DA ⊆ χθ . ∴

U{D : D ∈ IFS ( R) such that DB ⊆ χθ } ⊆ U{D : D ∈ IFS ( R) such that DA ⊆ χθ }

⇒ ( χθ : B ) ⊆ ( χθ : A).

Theorem (3.11) Let M is a R-module and A, B ∈ IFM(M), then

( χθ : A + B) = ( χθ : A) ∩ ( χθ : B).

Proof. Since A, B ∈ IFM(M) ⇒ A + B ∈ IFM(M), we have µ A+ B ( x) = ∨ {µ A ( y) ∧ µB ( z )} ≥ µ A ( x) ∧ µ B (θ ) = µ A ( x) and x= y+ z

ν A+ B ( x) = ∧ {ν A ( y ) ∨ ν B ( z )} ≤ ν A ( x) ∨ν B (θ ) = ν A ( x), ∀ x ∈ M . x= y+ z

This implies that A ⊆ A + B and B ⊆ A + B. So, ( χθ : A + B) ⊆ ( χθ : A) and ( χθ : A + B) ⊆ ( χθ : B) ⇒ ( χθ : A + B) ⊆ ( χθ : A) ∩ ( χθ : B).

Now, ( χθ : A) ∩ ( χθ : B) = ( ∪ { A1 | A1 ∈ IFI ( R), A1 A ⊆ χθ } ) ∩ ( ∪ { B1 | B1 ∈ IFI ( R), B1 B ⊆ χθ } ) = ∪ { A1 ∩ B1 | A1 , B1 ∈ IFI ( R), A1 A ⊆ χθ , B1 B ⊆ χθ } ⊆ ∪ {C | C = A1 ∩ B1 ∈ IFI ( R ), CA ⊆ χθ , C B ⊆ χθ } ⊆ ∪ {C | C = A1 ∩ B1 ∈ IFI ( R ), C ( A + B) ⊆ CA + CB ⊆ χθ } = ∪ {C | C ∈ IFI ( R ), C ( A + B) ⊆ χθ } = ( χθ : A + B).

Therefore, ( χθ : A) ∩ ( χθ : B) ⊆ ( χθ : A + B ).

Hence ( χθ : A + B) = ( χθ : A) ∩ ( χθ : B).

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International Journal of Computer Science & Information Technology (IJCSIT) Vol 9, No 4, August 2017

Theorem (3.12)   Let M is a R-module and Ai ∈IFM(M), i ∈J. Then  χθ : ∑ Ai  = I ( χθ : Ai ).  i∈J  i∈J

Proof. The proof is similar to above theorem (3.11).

4. Annihilator of intuitionistic fuzzy subset of a ring and module In this section we study annihilator of intuitionistic fuzzy subset of rings and modules in term of residual quotient and investigate various characteristic of it.

Definition (4.1) Let M be a R-module and A∈IFS(M), then the annihilator of A is denoted by ann(A) and is defined as: a n n ( A ) = U {B : B ∈ IF S ( R ) su ch th at B A ⊆ χ θ }.

Note that (χθ : A) = ann(A).

Lemma (4.2) Let M be a R-module, then ann( χθ ) = χ R .

Proof. The result follows from lemma (3.8) as ann( χθ ) = ( χθ : χθ ) = χ R .

Lemma (4.3) Let M be a R-module and A∈IFS(M), then χ 0 ⊆ ann( A).

Proof. The result follows from lemma (3.9) as χ 0 ⊆ ( χθ : A) = ann(A).

Lemma (4.4) Let M be a R-module and A, B∈IFS(M). If A ⊆ B, then ann ( B ) ⊆ ann ( A).

Proof. The result follows from lemma (3.10). If A ⊆ B then

ann(B) = ( χ θ : B ) ⊆ ( χθ : A) = ann(A).

Theorem (4.5) Let M be a R-module and A∈IFS(M). Then

ann ( A) = U {r(α , β ) : r ∈ R, α , β ∈ [0,1] with α + β ≤ 1 such that r(α , β ) A ⊆ χθ } Proof. The result follows from Theorem (3.2) Theorem (4.6) Let M be a R-module and A∈IFS(M). Then ann(A) A ⊆ χθ .

Proof. Follows from Theorem (3.3) (i) by taking A = χθ and B = A. Corollary (4.7) If A∈IFS(M) be such that µA(θ) = 1 and νA(θ) = 0, then ann( A) A = χθ .

Proof. By Lemma (4.3) we have χ 0 ⊆ ann( A)

⇒

µ χ (0 ) ≤ µ a n n ( A ) (0 ) a n d 0

i.e ., ⇒

1 ≤ µ a n n ( A ) (0 ) a n d

µ a n n ( A ) (0 ) = 1 a n d

ν χ (0 ) ≥ ν a n n ( A ) (0 )

0 ≥ ν

0

ann ( A )

(0 )

ν a n n ( A ) (0 ) = 0 .

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International Journal of Computer Science & Information Technology (IJCSIT) Vol 9, No 4, August 2017

Now, µ ann (A) A (θ ) = ∨ {µann (A) (r ) ∧ µ A (m) : r ∈ R, m ∈ M , rm = θ } ≥ µ ann (A) (0) ∧ µ A (θ ) = 1 ∧ 1 =1 i.e., µ ann (A) A (θ ) = 1. Similarly, we can show that ν ann (A) A (θ ) = 0.

Therefore, χθ ⊆ ann( A) A. Hence by Theorem (4.6) we get ann(A) A = χθ .

Note (4.8) If A∈IFM(M), then ann(A)A = χθ.

Theorem (4.9) Let M is a R-module and B ∈ IFS(R), A∈ IFS(M) such that BA⊆ χθ if and only if B ⊆ ann(A).

Proof. By definition of annihilator BA⊆ χθ ⇒ B ⊆ ann(A).

Conversely, let B ⊆ ann(A) ⇒ BA ⊆ ann(A)A ⊆ χθ .

Corollary (4.10) If in the above theorem (3.10) µB(0) = 1, νB(0) = 0 and µA(θ) = 1, νA(θ) = 0, then BA = χθ if and only if B ⊆ ann(A).

Theorem (4.11) Let M is a R-module and A, B∈IFS(M). Then the following conditions are equivalent:

(i)

ann( B) = ann( A), for all B ⊆ A, B ≠ χθ .

(ii ) CB ⊆ χθ implies CA ⊆ χθ , for all B ⊆ A, B ≠ χθ , C ∈ IFS ( R).

Proof. For (i) ⇒ (ii) Let CB ⊆ χθ. Then by theorem (3.11) we have C ⊆ ann(B) = ann(A) (by (i)). Again by the same theorem we have CA ⊆ χθ.

For (ii) ⇒ (i) By theorem (4.6) we have ann(B)B ⊆ χθ .

So (ii) implies ann(B)A ⊆ χθ where B ⊆ A, B ≠ χθ.

By theorem (4.8) ann(B) ⊆ ann(A).

Also, B ⊆ A ⇒ ann(A) ⊆ ann(B). Thus ann(A) = ann(B).

Corollary (4.12) If in the above theorem µA(θ) = 1, νA(θ) = 0 and µB(θ) = 1, νB(θ) = 0. Then the above theorem can be stated as: The following conditions are equivalents:

(i) ann(B) = ann( A), for all B ⊆ A, B ≠ χθ . (ii) CB = χθ implies CA = χθ , for all B ⊆ A, B ≠ χθ , C ∈ IFS (R) with µC (0) =1, ν C (0) =0.

Theorem (4.13) Let M is a R-module and A∈IFS(M). Then ann( A) = U{B: B∈ IFI (R) such that BA ⊆ χθ }, where IFI(R) is the set of intuitionistic fuzzy ideals of R. Proof. It follows from Theorem (3.5)(i)

Theorem (4.14) Let M is a R-module and A∈IFS(M). Then ann(A) ∈IFI(R).

Proof. It follows from Theorem (3.6) 11

International Journal of Computer Science & Information Technology (IJCSIT) Vol 9, No 4, August 2017

Theorem (4.15) Let M is a R-module and Ai∈IFS(M), i ∈ Λ. Then   ann  U Ai  = I ann( Ai ).  i∈Λ  i∈Λ Proof. It follows from Theorem (3.7)(i) Theorem (4.16) Let M is a R-module and Ai ∈ IFM(M) for i ∈J , then   ann  ∑ Ai  = I ann( Ai ).  i∈J  i∈J Proof. It follows from Theorem (3.12) Definition (4.17) Let M be R-module. Then A∈ IFS(M) is said to be faithful if ann(A) = χ0.

Lemma (4.18) Let A∈ IFS(M) be faithful, where M is R-module. If R is non-zero then

A

≠ χθ.

Proof. Since A is faithful ⇒ ann(A) = χ0.

If A = χθ then ann(A) = ann(χθ) = χR. Thus we have χ0 = χR ⇒ R = {0}, a contradiction.

Therefore, A ≠ χθ.

Theorem (4.19) Let A ∈ IFS(R) with µA(0) = 1, νA(0) = 0. Then A ⊆ ann(ann(A)) and ann (ann(ann(A))) = ann(A).

Proof. Let A be an intuitionistic fuzzy subset of R-module R. Then by corollary (4.7), we have ann(A)A = χ0.

By theorem (4.9), we have A ⊆ ann(ann(A)) ...............(1) ⇒ ann(ann(ann(A))) ⊆ ann(A) [ using lemma (4.4) ] Again using (1) : ann(A) ⊆ ann(ann(ann(A))).

So, ann (ann(ann(A))) = ann(A).

Theorem(3.20) Let A ∈ IFS(M). Then C(α , β ) ( ann( A)) ⊆ ann(C(α , β ) ( A)), ∀ α , β ∈ (0,1] with α + β ≤ 1.

Proof. Let x ∈ C(α , β ) (ann( A)). Then µann ( A) ( x) ≥ α > 0 and ν ann ( A) ( x) ≤ β < 1 ⇒ ∨ {µ B ( x) : B ∈ IFI ( R), BA ⊆ χθ } ≥ α and ∧ {ν B ( x) : B ∈ IFI ( R), BA ⊆ χθ } ≤ β ⇒ µ B ( x) ≥ α and ν B ( x) ≤ β for some B ∈ IFI ( R) with BA ⊆ χθ .

If x ∉ ann(C(α , β ) ( A)) then ∃'s some y ∈ C(α , β ) ( A) such that xy ≠ θ .

Now, µ BA ( xy ) ≥ µ B ( x) ∧ µ A ( y ) ≥ α > 0 and ν BA ( xy ) ≤ ν B ( x) ∨ ν A ( y ) ≤ β < 1, which is a contradiction. Hence C(α , β ) (ann( A)) ⊆ ann(C(α , β ) ( A)).

Definition (4.21) A∈IFI(R) is said to be an intuitionistic fuzzy dense ideal if ann(A) = χ0 .

Definition (4.22) A∈IFI(R) is called intuitionistic fuzzy semiprime ideal of R if for any IFI B of R such that B2 ⊆ A implies that B ⊆ A. 12

International Journal of Computer Science & Information Technology (IJCSIT) Vol 9, No 4, August 2017

Theorem (4.23) If A is an IFI of a semi prime ring R, then A ∩ ann(A) = χ0 and A + ann(A) is an intuitionistic fuzzy dense ideal of R.

Proof. Since A ∩ ann(A) ⊆ A , A ∩ ann(A) ⊆ ann(A) so (A ∩ ann(A))2 ⊆ A ann(A) ⊆ χ0.

Now R is a semiprime ring and it implies that 0 is a semiprime ideal of R so χ0 is an intuitionistic fuzzy semi prime ideal of R. Also, (A ∩ ann(A))2 ⊆ χ0 ⇒ A ∩ ann(A) ⊆ χ0 and hence A ∩ ann(A) = χ0 .

Hence ann(A + ann(A)) = ann(A) ∩ ann(ann(A)) = χ0 proving thereby A+ ann(A) is an intuitionistic fuzzy dense ideal of R.

Theorem (4.24) Let A be a non-zero intuitionistic fuzzy ideal of a prime ring R with µA(0) = 1, νA(0) = 0. Then A is an intuitionistic fuzzy dense ideal of R. Proof. Now, A ann(A) = χ0 ⇒ ann(A) = χ0 or A = χ0. But A ≠ χ0 so ann(A) = χ0.

Hence A is an intuitionistic fuzzy dense ideal of R.

Definition (4.25) If A∈ IFS(R). Then the intuitionistic fuzzy ideal of the form ann(A) is called an intuitionistic fuzzy ideal. Thus if A is an intuitionistic fuzzy annihilator ideal if and only if A = ann(B) for some B ∈IFS(R) with with µB(0) = 1, νB(0) = 0.

Remark (4.26) In view of theorem (4.19) it follows that A is an annihilator ideal of R implies ann(ann(A)) = A.

Theorem (4.27) The annihilator ideals in a semiprime ring form a complete Boolean algebra with intersection as infimum and ann as complementation.

Proof. Since





I ann( A ) = ann  ∑ A  , so any intersection of annihilator ideals is an i

i∈I

i

i∈I

intuitionistic fuzzy annihilator ideal. Hence these ideals form a complete semi-lattice with intersection as infimum. To show that they form a Boolean algebra it remain to show that: A ∩ ann(B) = χ 0 if and only if A ⊆ B, for any annihilator ideals A and B. If A ⊆ B then A ∩ ann(B) ⊆ B ∩ ann(B) = χ 0 .

Conversary, let A ∩ ann(B) = χ 0 .

Now, A ann(B) ⊆ A ∩ ann(B) = χ 0 ⇒ A ⊆ ann( ann(B)) = B.

Theorem (4.28) Let M be a non-zero R-module. Suppose that there exist no ideal A maximal among the annihilators of non-zero intuitionistic fuzzy submodules (IFSMs) of M. Then A is an intuitionistic fuzzy prime ideal of R.

Proof. Since A is maximal among the annihilators of non-zero intuitionistic fuzzy submodules (IFSMs) of M. Therefore there is an IFSM B (≠ χ0) of M such that A = ann(B).

Suppose P, Q ∈ IFI(R) properly containing A ( i.e., A ⊂ P and A ⊂ Q ) such that PQ ⊆ A. If QB = χ0 then Q ⊆ ann(B) = A, which is a contradiction to our supposition so QB ≠ χ0.

Now, PQ ⊆ A ⇒ P(QB) ⊆ AB = ann(B)B = QB = χ0 . So Q ⊆ ann(QB).

Hence A ⊆ ann(QB). This is a contradiction of the maximality of A. So A is an intuitionistic fuzzy prime ideal of R.

Remark (4.29) If A∈IFM(M) , A ≠ χθ satisfying one (hence both) the condition of Theorem (4.11) then A is called an intuitionistic fuzzy prime submodule of M. 13

International Journal of Computer Science & Information Technology (IJCSIT) Vol 9, No 4, August 2017

Theorem (4.30) If A is an intuitionistic fuzzy prime submodule of M then ann(A) is an intuitionistic fuzzy prime ideal of R.

Proof. Let A be an intuitionistic fuzzy prime submodule of M and PQ ⊆ ann(A) , where Q is not contained in ann(A). Then χθ ≠ QA ⊆ A. Now PQ ⊆ ann(A) ⇒ (PQ)A ⊆ ann(A)A = χθ .

So, P ⊆ ann(QA) = ann(A), as A is prime. Hence ann(A) is prime ideal of R.

5. CONCLUSIONS In this paper we have developed the concept of residual quotient of intuitionistic fuzzy subset of rings and modules and then study some results on annihilator of an intuitionistic fuzzy subset of a R-module. Using this notion, we investigate some important characterization of intuitionistic fuzzy annihilator of subsets of modules. The annihilator of union (sum) of intuitionistic fuzzy submodules are obtained. Annihilator of intuitionistic fuzzy ideal of prime ring, semi prime ring are also obtained. Using the concept of intuitionistic fuzzy annihilators, intuitionistic fuzzy prime submodules and intuitionistic fuzzy ideals are defined and various related properties are established.

ACKNOWLEDGEMENTS The second author would like to thank IKG PT University, Jalandhar for providing the opportunity to do research work.

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D. S. Malik and J. N. Mordeson, (1998), Fuzzy Commutative Algebra, World Scientific Publishing Co-Pvt. Ltd. K. Meena and K. V. Thomas, (2011), Intuitionistic L-Fuzzy Subrings, International Mathematical Forum, Vol. 6, No. 52, pp 2561 – 2572.

P.K. Sharma, (2011), (α, β)-Cut of Intuitionistic fuzzy modules- II, International Journal of Mathematical Sciences and Applications, Vol. 3 , No. 1, pp. 11-17.

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Authors Dr. P.K. Sharma, Ph.D. is Associate Professor in Mathematics at D.A.V. College, Jalandhar, Punjab, India. Having a teaching experience of 24 years. His area of research include Fuzzy Algebra, Fuzzy Topology, Fuzzy Graph Theory, Intuitionistic fuzzy mathematics. Till date he has published 100 research papers in reputed National and International Journals. He has written four post-Graduate books on Algebra, Topology, Real Analysis and Discrete Mathematics.

Ms. Gagandeep Kaur, presently p ursing Ph.D from IKGPTU, Jalandhar, Punjab, India. She has a teaching experience of 4 years in CT group of Institutes, Shahpur, Jalandhar as Assistant professor in Mathematics.

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