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Arvind Kumar Sinha, Manoj Kumar Dewangan / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 4, Jul-Aug 2013, pp.852-854

Isomorphism Theorems for Fuzzy Submodules of G-Modules Arvind Kumar Sinha, Manoj Kumar Dewangan Department of Mathematics, NIT, Raipur, Chhattisgarh, India

ABSTRACT In this paper we give isomorphism theorems for fuzzy submodules of G-modules. MATHEMATICAL SUBJECT CLASSIFICATION 03E72 KEY WORDS: Fuzzy G-submodule, Quotient Gmodules, Fuzzy isomorphism theorem.

Further in above definition if

 is one-one

 is an isomorphism. Definition-2.3[5]: Let  : M  M  be a Gmodule homomorphism. The set of all m  M such that  (m)  0 ,the zero element of M  is called the kernel of  and is denoted by ker  and onto, then

In other words

I.

The theory of group representation (Gmodule theory) was developed by Frobenious G[1962]. Soon after the concept of fuzzy sets was introduced by Zadeh [1] in 1965.Fuzzy subgroup and its important properties were defined and established by Rosenfeld [2] in 1971.After that in the year 2004 Shery Fernandez [3] introduced fuzzy parallels of the notions of G-modules. In this paper we give isomorphism theorems for fuzzy submodules of G-modules.The idea came for proving these results from Mordeson and Malik [6] which was originally proved for Lsubmodules of R-modules M. Throughout the paper we use the notation  for homomorphism,  for isomorphism.

II.

 i  m.1G  mm  M  ii  m.( g.h)  (m.g ).h

Definition-2.5[5]: If M is a G-module and N is a Gsubmodule of M, then M/N is a G-module which is called Quotient G-modules. Let g  G and x  N 

M . Define the N

action of G on M/N by

g ( x  N )  gx  N 

M N

Definition-2.6[5]: Let G be a finite group and M be a G-module over K, Which is a subfield of C. Then a fuzzy G-module on M is a fuzzy subset  of M such that

(i )  (ax  by )   ( x)   ( y )a, b  K , x, y  M . (ii )  ( gm)   (m)g  G, m  M .

Definition-2.7[6]:

(iii )(k1m1  k2 m2 ).g  k1 (m1.g )  k2 (m2 .g ) k1 , k2  K , m, m1, m2  M , g , h  G

1G being identity element in G.

Definition-2.2[5]: Let M and mapping  : M  M  is homomorphism if

Definition-2.4[4]: Let M be a G-module. A vector subspace N of M is a G-submodule if N is also a Gmodule under the same action of G.

This satisfies all the condition of G-module and therefore M/N is a G-module.

PRELIMINARIES

To prove isomorphism theorems for fuzzy submodules of G-modules,we use the following definitions. Definition 2.1 [5]: Let G be a finite group. A vector space M over a field K is called a G-module if for every g  G and m  M,there exist a product (called the action of G on M) m.g  M satisfying the following axioms :

Where

ker   {m  M :  (m)  0}

INTRODUCTION

M ' be G-modules. A a

G-module

(i ) (k1m1  k2 m2 )  k1 (m1 )  k2 (m2 ) (ii ) ( g.m)  g. (m)

 G

M

Let M

(Where G denotes the fuzzy power set of G-module M).Then  is called a fuzzy submodule of G-module M, if

(i)  (0)  1G

(ii)  ( gm)   (m)g  G, m  M (iii)  (m1  m2 )   (m1 )   (m2 )m1 , m2  M Definition-2.8[6]: Let   G ( M ) (Where G (M) denotes the set of all fuzy submodules of G-module M) and let N be a submodule of M. Define

k1 , k2  K , m, m1, m2  M , g  G 852 | P a g e


Arvind Kumar Sinha, Manoj Kumar Dewangan / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 4, Jul-Aug 2013, pp.852-854 M

M

 G N

(Where G N is the fuzzy power set of Gmodule M/N) as follows:

 ([ x])  { (u) : u [ x]} x  M

 {{ ( y) : y  x }: x   , g ( x)  z}  { ( y) : y   , g ( y)  z}  { ( y ) : y  G, f ( y )  z}

  ( z ), z   

Where M/N denotes the quotient module of M w.r.t. N and [x] represents the coset x + N. Then   G (M/N) Definition-2.9[6]:

  G( M )

If

then

  {x  X :  ( x) 0} is called support of  .

 h . 

III.

MAIN RESULTS

We give the following results. Theorem-3.1 (First isomorphism theorem) Let   G ( M ) and   G( N ) such that   .Then there exists

  G( M ) such that

   and   .  Proof Since    there exists an of M onto N such that f ( )   . M Define   G as follows

 ( x)  

Theorem-3.2 (Second isomorphism theorem) Let  , be fuzzy submodules of G-module M with  (0)   (0) then

 (   )

  G( M )

then If x  ker f

(     )

epimorphism f

(    )      

0, xker f

then

 ( x)  0

 ( yxy )   ( x)  ( y ) . Hence  is normal G-submodule

Consequently we have

of

.Also

f

   f ( )   .

g

f |

( f ( ))   

f (  )    .

then g is a homomorphism of

  onto   and kerg =   by definition of  .Then there exists an isomorphism h of   /   onto

Where f is given by

f ( x(   ) )  x  , x           f  ( y )    ( y(  ) ) (    ) (    )     (Since f is one-one)

 { ( z ) : z  y (   )}

It follows that Let

f  (    ) (   )    

Thus and so

1

Which further implies

    

(   )  (     )

yxy 1  ker f , y  M .

x  G \ ker f

One can verify that

 ( yxy 1 )   ( yxy 1 )   ( x)  ( y )   ( x)  ( y ) If



  .

and

(   )

Proof We have is normal G-submodule of M. By the second isomorphism theorem for module

 ( x ), xker f

Then

such that

h( x  )  g ( x)  f ( x)x   For such an h, we have

h( /  )( z )  {( /  )( x  ) : x   , h( x  )  z}

 {(   )( z ) : z  y (     )  {(   )( z ) : z  y  }  (   )      ( y  ), y     Hence

   (   ) f    (   )   (   )   (   )  Theorem-3.3 (Third isomorphism Theorem)

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Arvind Kumar Sinha, Manoj Kumar Dewangan / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 4, Jul-Aug 2013, pp.852-854 Let  , , 

be fuzzy submodules of G-module M

     then ( /  )   . ( /  )   Proof Since      then  is submodule of   and both   and   are submodule of   .By the with  (0)   (0) and

third isomorphism theorem for modules

(  /   ) f    (  /   )   Where f is defined by

f ( x  (  /   ))  x  , x    Thus

 ( /  )    ( /  )     f  ( x )    ( x ( /  )) (  /  ) (  /  )        {( /  )( y ) : y   , y   x  (  /   )}  {{ ( z ) : z  y }: y    , y   x  (  /   )}

 { ( z ) : z    , f ( z )  x }      ( x  )   Hence

( /  ) f  . ( /  )   REFERENCES [1] [2] [3]

[4] [5]

[6]

L.A Zadeh.,Fuzzy sets, J.Inf. Control, 8(1965)383-353. A.Rosenfeld,Fuzzy groups, J.Math. Anal. Appl. 35(1971) 512-517. Shery Fernandez, Fuzzy G-modules and fuzzy representation, TAJOPAAM, I(2002) 107-114. Shery Fernandez, Ph.D. thesis “A study of fuzzy g-modules” April 2004. Charles. W. Curvies and Irving Reiner, Representation Theory of finite group and Associative algebra.INC,(1962). J. N. Mordeson and D. S Malik., Fuzzy Commutative Algebra, World scientific publishing, (1998).

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