 Open Access
 Total Downloads : 281
 Authors : B. Suganya, Dr. N. Sarala
 Paper ID : IJERTV4IS020765
 Volume & Issue : Volume 04, Issue 02 (February 2015)
 Published (First Online): 27022015
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Q Fuzzy Soft Ring
Dr. N. Sarala , B. Suganya
Department of Mathematics, Department of Mathematics,

college for women(Autonomous), Bharathidasan University constituent College of Nagapattinam, TamilNadu,(India) Arts and science,
Nagapattinam, Tamilnadu(India)
Abstract: – In this paper, the study of Q fuzzy soft ring by combining soft set theory. The notions of Q fuzzy soft ring as defined and several related properties and structural characteristics are investigated some related properties., then the definition of Q fuzzy soft ring and the theorem of homomorphic image and homomorphic preimage are given.
Keywords: Soft set, Fuzzy soft set, soft ring, Fuzzy soft ring, soft homomorphism, Fuzzy Soft isomorphism,Qfuzzy set , Q Fuzzy soft ring

INTRODUCTION
The concept of soft sets was introduced by Molodtsov [3] in 1999, soft sets theory has been extensively studied by many authors. It is well known that the concept of fuzzy sets, introduced by Zadeh [9], has been extensively applied to many scientific fields.
In 1971, Rosenfeld [4] applied the concept to the theory of groupoids and groups. Inan et al. have already introduced the definition of fuzzy soft rings and studied some of their basic properties. A.Solairaju and R.Nagarajan[7] have introduced and defined a new algebraic structure called Qfuzzy subgroups. (Dr.N.sarala and B.Suganya, 2014) presented some properties of fuzzy soft groups . further (Dr.N.sarala and B.Suganya, 2014) introduced on normal fuzzy soft groups.
In this paper, we study Qfuzzy soft ring theory by using fuzzy soft sets and studied some of algebraic properties.

PRELIMINARIES
In this section, we first recall the basic definitions related to fuzzy soft sets which would be used in the sequel.
Definition2.1
Suppose that U is an initial universe set and E is a set of parameters, let P(U) denotes the power set of U .A pair (F,E) is called a soft set over U where F is a mapping given by F: E P(U) .
Clearly, a soft set is a mapping from parameters to P(U), and it is not a set, but a parameterized family of subsets of the Universe.
Definition 2.2.
Let U be an initial Universe set and E be the set of parameters. Let A E. A pair (F, A) is called fuzzy soft set over U where F is a mapping given by F: AIU, where IU denotes the collection of all fuzzy subsets of U.
Definition 2.3
Let X be a group and (F,A) be a soft set over X. Then (F,A) is said to be a soft group over X iff F(a) < X, for each a A.
Definition 2.4
Let X be a group and (f, A) be a fuzzy soft set over X. Then (f, A) is said to be a fuzzy soft group over X iff for each a A and x, y X,

fa(x . y) T (fa(x), fa(y)

fa(x1) fa(x)
That is, for each a A, fa is a fuzzy subgroup in Rosenfeld's sense [4]
Definition 2.5
Let (f,A) be a soft set over a ring R. Then (f, A) is said to be a soft ring over R if and only if f(a) is sub ring of R for each a A.
Definition 2.6
Let R be a soft ring. A fuzzy set in R is called
fuzzy soft ring in R if
(i) ((x+y)) T{(x), (y)}

(x) (x) and

((xy)) T{(x), (y)} , for all x,y R.
Definition 2.7
Let (,): X Y is a fuzzy soft function, if is a homomorphism from x y then (,) is said to be fuzzy soft homomorphism. if is a isomorphism from X Y and is 11 mapping from A on to B then (,) is said to be fuzzy soft isomorphism.


QFUZZY SOFT RINGS

Definition 3.1:
Let R be a soft ring. A fuzzy set in R is called
Q fuzzy soft ring in R if
(i) ((x+y) , q) T{(x,q), (y,q)}

(x, q) (x,q) and

((xy) , q) T{(x,q), (y,q)}, for all x,y R.
Proposition 3.1:
Every imaginable Q fuzzy soft ring is a Q fuzzy soft ring of R.
Proof:
Assume that is imaginable Q fuzzy soft ring of R, then we have
((x+y) , q) T{(x,q), (y,q)} (x, q) (x,q) and
((xy) , q) T{(x,q), (y,q)} , for all x,y R.
Since is imaginable, we have min{(x,q) ,(y,q)} = T{min{(x,q), (y,q)},
min{x,q) , (y,q) }}
T {(x,q) , (y,q)}
min {(x,q) , (y,q)}
Proposition 3.3:
Let R and R' be two rings and : R R' be a soft homomorphism. If and fa is a Qfuzzy soft ring of R then the preimage 1 (fa) Qfuzzy soft ring of R.
Proof:
Assume that fa is a Qfuzzy soft ring of R'. Let x, y R & q Q
(FSR1)
(i) ((x+y),q)) = ((x+y),q))
1[fa] fa
= ((x,q ), (y,q)) fa
T { ((x,q)), ((y,q))} fa fa
T { (x,q), (y,q)}
1[fa] 1[fa]
(FSR2)
(ii) (x,q) = ((x,q))
1[fa] fa
((x,q) fa
and so
T{(x,q) ,(y,q)} = min { (x,q) , (y,q) }
It follows that
((x+y), q ) T{(x,q) , (y,q)}
(FSR3)
(x,q)
1[fa]
x,y R, qQ
= min {(x,q) , (y,q)} for all
(iii) ((xy),q)) = ((xy),q))
1[fa] fa
Hence is a Qfuzzy soft ring of R.
Proposition 3.2:
If is Qfuzzy soft ring R and is an endomorphism of R, then [] is a Q Fuzzy soft ring of R
Proof:
For any x,y R, we have
(FSR1)
(i) []((x+y),q)) = (((x+y),q))
= ((x,q ),(y,q))
T { ((x,q)),((y,q))}
T { [] (x,q),[](y,q)}
(FSR2)
(ii) [](x,q) = ((x,q))
((x,q))
[](x,q)
(FSR3)
(iii) []((xy),q)) = (((xy),q))
= ((x,q ), (y,q))
T {(x,q) , (y,q )}
T { (x,q) , (y,q )}
T { [] (x,q),[](y,q)} Hence [] is a Qfuzzy soft ring of R.
= ((x,q ), (y,q)) fa
T { ((x,q)), ((y,q))} fa fa
T { (x,q), (y,q) }
1[fa] 1[fa] Hence 1 (fa) is a Qfuzzy soft ring of R.
Proposition 3. 4
Let: R R' be an epimorphism and fa be fuzzy soft set in R' . If [fa] is qfuzzy soft ring of R then fa is qfuzzy soft ring of R.
Proof :
Let x,y R, Then there exist a,b R such that
(a) = x , (b) =y. It follows that
(FSR1)
(i) ((x+y),q)) = ((x+y),q))
[fa] fa
(FSR2)
= ((x,q ), (y,q)) fa
T { ((x,q)), ((y,q))} fa fa
T { (x,q), (y,q)}
[fa] [fa]
(FSR1)
(i) f((x1+y1), q) = sup (z,q)
(z,q) f1((x1+y1),q)
T{(x0,q), (y0,q)
= T{sup (h,q), sup (h,q)} (h,q) f1(x1,q) , (h,q) f1(y1,q)
= min {f(x1,q) , f (y1,q)}
(ii) (x,q) = ((x,q))
[fa] fa((x,q)) fa
(x,q)
[fa](FSR3)
(iii) ((xy),q)) = ((xy),q))
[fa] fa= ((x,q ), (y,q)) fa
T { ((x,q)), ((y,q))} fa fa
T { (x,q), (y,q)}
[fa] [fa] Hence [fa] is a Qfuzzy soft ring of R.Proposition 3.5:
Onto homomorphic image of a Qfuzzy soft ring with the sup property is Qfuzzy soft ring of R.
Proof:
Let f: RR' be an onto homomorphism of Q fuzzy soft rings and let be a sup property of Qfuzzy soft ring of R.
Let x1, y1 R1, and x0 f1(x1), y0 f1(y1) be such that (x 0,q) = sup (h,q),
(h,q) f1(x1) and
(y 0,q) = sup (h,q)
(h,q) f1(y1) Respectively, then we can deduce that
(FSR2)

f(x1,q) = sup (z,q)
(z,q) f1( x1,q)
(x0 , q)
sup (,q) (h,q) f1(x1,q)
= f(x1,q)
(FSR3)

f((x1y1), q) = sup (z,q)
(z,q) f1((x1y1),q)
T{(x0,q), (y0,q)}
= T { sup (h,q) , sup (h,q)} (h,q) f1(x1,q) , (h,q) f1(y1,q)
= min {f(x1,q) , f (y1,q)} Hence f is a Qfuzzy soft ring of R1.
Proposition 3. 6:
Let T be a continuous tnorm and Let f be a soft homomorphism on R. If is Qfuzzy soft of R, then f is Qfuzzy soft ring of f(R).
Proof:
Let A1 = f 1(y1,q) , A2 = f 1(y2,q) and A12 = f 1((y1+y2), q) where y1,y2 f(R), q Q
Consider the set
A1+ A2 = { x R / (x,q) = (a1,q) + (a2,q) } for some (a1,q)A1 and (a2,q) A2.
If (x,q) A1+A2 , then (x,q) = (x1,q) + (x2,q) for some (x1,q) A1 and (x2,q) A2 so that we have
f (x,q) = f(x1,q)+ f(x2,q)
= y1+ y2
Since (x,q) f1((y1,q)+ (y2,q)) = A12 .Thus A1+A2
A12
It follows that
(FSR1)
(i) f ((y1+y2), q ) = sup{(x,q)/(x,q)
f1(y1+ y2,q)}
= sup {(x,q) / (x,q) A12 }
sup {(x,q)/ (x,q) A1+A2}
sup {((x1,q)+(x2,q))/(x1,q) A1 and
(x2,q) A2}
sup {S((x1,q) , (x2,q))/ (x1,q) A1
and (x2,q) A2}
Since T is continuous. For every > 0 , we see that if
sup {(x1,q) / (x1,q)A1} + (x1*, q) } and
sup {(x2,q) / (x2,q) A2} + (x2*,q)}
T{sup{(x1,q)/ (x1,q)A1} ,
sup { (x2,q) / (x2,q)A2 } +T((x1*,q), (x2*,q)
(FSR2)
(FSR3)
*(x,q) = (x,q) +1 (0,q)
(x,q) +1 (0,q)
= (x,q)
*((xy),q) = ((xy),q) + 1 (0,q)
T((x,q), (y,q)) +1 (0,q))
T((x,q)+1 (0,q),
((y,q)+1 (0,q))
= T (*(mx,q) , *(my,q)).
Choose (a ,q) A and (a ,q) A
such that
3. CONCLUSION:
1 1 2 2
sup{(x1,q) / (x1,q) A1} + (a1,q) and
sup{(x2,q) / (x2,q) A2} +(a2,q) .
Then we have
T{sup{(x1,q)/(x1,q)A1}, sup{(x2,q)/(x2,q)
A2}+T((a1,q),(ma2,q)
Consequently, we have
f((y1+ y2), q ) sup{ T((x1,q), (x2,q))
/ (x1,q) A1 ,(x2,q) A2}
T(sup{(x1,q) / (x1,q) A1},
sup{(x2,q) / (x2,q) A2}
T {(f(y1,q) , f(y2,q)}
Similarly we can show f(x,q ) f(x,q) and f(xy,q) T {(f(x,q) , f(y,q)}
Hence f is Qfuzzy soft ring of f(R).
Proposition3.7:
Let be a Qfuzzy soft ring R and let * be a Q fuzzy set in N defined by *(x,q) = (x,q) +1(0,q) for all x N. Then * is a normal Qfuzzy subgroup of R
Proof :
In this paper we investigate the notion of Qfuzzy soft ring. This work focused on Qfuzzy soft rings of fuzzy soft rings. To extend this work one could study the properties of fuzzy soft sets in other algebraic structure.
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For any x, yR and q Q we have
(FSR1)
*((x+y),q) = ((x+y),q) + 1 (0,q)
T((x,q), (y,q)) +1 (0,q))
T((x,q)+1(0,q),
((y,q)+1(0,q))
= T (*(mx,q) , *(my,q)).