Fuzzy Soft Rings on Fuzzy Lattices

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1 International Journal of Computational Science and Mathematics. ISSN Volume 3, Number 2 (20), pp International Research Publication House Fuzzy Soft Rings on Fuzzy Lattices M. Marudai and V. Rajendran 2 Associate Professor, Research Scholar, Department of Mathematics Bharathidasan University, Tiruchirappalli, Tamilnadu, India marudaim@hotmail.com, rajendranmanohari@yahoo.in Abstract Molodtsov [999] initiated the concept of soft set theory as a new approach for modeling uncertainities. Then Maji et. al [200] expanded this theory to fuzzy soft set theory. The algebraic structures of soft set theory have been studied increasingly in recent years. Aktas and Ca gman [2007] defined the notion of soft groups. Feng et.al [2008] initiated the study of soft semi rings and finally soft rings are defined by Acar et.al [200]. In this study we introduce, fuzzy soft ring, which is a generalisation of soft rings introduced by Acar et. al. and we study some of their properties. Keywords: Soft ring, Fuzzy Soft ring, imaginable, pre-image, soft homomorphism, fuzzy soft isomorphism. Introduction Most of the existing mathematical tools for formal modeling, reasoning and computing are crisp, deterministic and precise in character. But in real life situation, the problems in economics, engineering, environment, social science, medical science etc do not always involve crisp data. Consequently, we cannot successfully by using the traditional classical methods because of various types of uncertainties in this problem. There are several theories, for example, theory of fuzzy sets [27], theory of intuitionist fuzzy sets [5], vague sets [3], interval mathematics [26], and rough sets [23], which can be considered as mathematical tools for dealing with uncertainties. But all these theories have this inherents difficulties as what where point out by Molodtsov in [2]. The reason for these difficulties is possibly the inadequacy of the parameterization tool of the theories. In 999 Molodtsov [2], initiated the Novel concept of soft set theory which is completely new approach for modeling vagueness and uncertainties. Soft set theory has a rich potential for applications in several directions, few of which had been

2 42 M. Marudai and V. Rajendran shown by Molodtsov in [2] After Molodtsov work, some different applications of soft set theory where studied in []. Furthermore Maji, Biswas and Roy worked on soft set theory in [8]. Also Maji et. al [7] presented the definition of fuzzy soft set and Roy et. al presented some applications of there notion to decision making problems. The algebraic structures of set theories dealing with uncertainities has also been studied by some authors. Rosenfield [25] proposed the concept of fuzzy groups in order to establish the algebraic structures of fuzzy sets. Rough groups were defined by Biswas and some authors have studied the algebraic properties of rough sets as well. Recently the many authors discuss the soft set research on the soft set theory is progressing rapidly. For example, the concept of soft semi ring, soft group, soft BCK/BCI algebras, soft BLalgebras and fuzzy soft groups. This paper begins by introducing the basic concept of fuzzy soft set theory, then we introduce the basic version of fuzzy soft ring theory, which extends the notion of the ring to the algebraic structure of fuzzy soft set. In this paper, we study Molodtsov motion of soft sets and fuzzy soft set considering the fact that the parameters are mostly fuzzy hedges or fuzzy parameters. We discuss fuzzy soft sets algebraic structure and given the definition of fuzzy soft ring. We define operations on fuzzy soft rings soft set, research on the soft set theory is progressing rapidly. For example, the concept of soft semi ring, soft group, soft BCK/BCI algebras, soft BL-algebras and fuzzy soft groups. This paper begins by introducing the basic concept of fuzzy soft set theory, then we introduce the basic version of fuzzy soft ring theory, which extends the notion of the ring to the algebraic structure of fuzzy soft set. In this paper, we study Molodtsov motion of soft sets and fuzzy soft set considering the fact that the parameters are mostly fuzzy hedges or fuzzy parameters. We discuss fuzzy soft set algebraic structure and given the definition of fuzzy soft ring. We define operations on fuzzy soft rings and prove some results on them. Finally we present image, pre-image, fuzzy soft homomorphisms and discussed their properties. Preliminaries Through out this paper R denotes a commutative ring and all fuzzy soft sets are considered over R. Definition 2. A pair (f, A), is called a soft set over the lattice L, If f:a P(L). Here L be the initial universe and E be the set of parameters. Let P(L) denotes the power set of L and I L denotes the set of all fuzzy sets on L. Definition 2.2 A pair (f, A is called a fuzzy soft set over L, where f:a I L, ie. for each a A, f a : L I is a fuzzy set in L.

3 Fuzzy Soft Rings on Fuzzy Lattices 43 Definition 2.3 Let (f, A) be a non-null 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 2.4 Let (f, A) be a non Null fuzzy soft set over a ring R. Then (f, A) is called a fuzzy soft ring over R If and only if f(a) = s a fuzzy sub ring of R. for each a A. (FSR) f a (x + y) T {f a (x), f a (y)} (FSR2) f a ( x) f a (x) f a (xy) T{ f a (x), f a (y)} for all x, y R Definition 2.5 Let (f, A) be a fuzzy soft set over L. The soft set (f, A) α = {(f a ) α / a A} for each α [0, ] is called α level soft set. Definition 2.6 Let f a be a fuzzy soft ring in R. Let θ:r R be a map and define f θ a (x) = f a (θx) by defining f θ a : R [0, ]. Definition 2.7 Let φ: X Y and Ψ: A B be two functions, where A and B are parameter sets for the crisp sets X and Y respectively. Then the pair (φ, Ψ) is called a fuzzy soft function from X to Y. Definition 2.8 The pre-image of (g, B) under the fuzzy soft function (φ, Ψ) denoted by (φ, Ψ) is the fuzzy soft set defined by (φ, Ψ) (g, B) = (φ (g), Ψ (B)). Definition 2.9 Let (φ, Ψ): X Y is a fuzzy soft function, if φ is a homomorphism from X Y then (φ, Ψ) is said to be fuzzy soft homomorphism. If φ is a isomorphism from X Y and Ψ is mapping from A on to B then (φ, Ψ) is said to be fuzzy soft isomorphism. Properties of Fuzzy Soft Ring Preposition 3. Let f a be a fuzzy soft ring of R then f a (x) f a (0) for all x R, the subset Rf a = {x R / f a (x) = f a (0)} is a fuzzy soft ring of R.

4 44 M. Marudai and V. Rajendran Let x G, then (FSR) f a (x + y) = T { f a (x), f a (y)} = T { f a (x), f a ( x)} f a ((x) + ( x)) f a (0) This implies subdivision (i). To verify subdivision (ii), it follows that x R f a and R f a φ. Now let x, y R, then (FSR2) fa(x + ( y)) T{f a (x), f a ( y)} = T {f a (x), f a (y)} = T {f a (0), f a (0)} = f a (0) But, from subdivision (i), f a (x) f a (0) for all x, y R and so f a (x + ( y)) = f a (0). Which then x + ( y) Rs fuzzy soft ring of R. Let x, y Rf a and setting y = x, then f a (xy) T{f a (x), f a (y)} = T {f a (x), f a (x)} = T {f a (0), f a (0)} = f a (0) s a fuzzy soft ring over R. Corollary: 3.2 Let R be a finite ring and f a be a fuzzy soft ring of R. Consider the subset H of R given by H = {x R / f a (x) = f a (0) } Then H is called crisp subring of R. Proposition 3.3 Let f a and f b be two fuzzy soft rings of R then f a f b is fuzzy soft ring of R. Let x, y R (FSR) (f a f b ) (x + y) = T {f a (x + y), f b (x + y)} T {T f a (x), f a (y)}, T{ f b (x), f b (y) } T {T { f a (x), f a (y), f b (x), f b (y)}}

5 Fuzzy Soft Rings on Fuzzy Lattices 45 T {T { f a (x), f b (y)}, T{f a (y), f b (y)}} T {(f a f b )(x), (f a f b )(y) } FSR satisfied in R (FSR2) (f a f b ) ( x) = T {f a ( x), f b ( x)} T {f a (x), f b (x)} {(f a f b )(x) (f a f b ) (xy) = T {f a (xy), f b (xy)} T { T {f a (x), f a (y)}, T{ f b (x), f b (y)}} T { T {f a (x), f a (y)}, f b (x), f b (y)}} T { T {f a (x), f b (x)}, T{ f a (y), f b (y)}} T { (f a f b ) (x), (f a f b ) (y)} Hence f a f b is fuzzy soft ring of R Preposition 3.4 If s fuzzy soft ring of R then the non-empty level subset U (f a ; t) is fuzzy soft ring for all t I m (f a ). Let f a be a fuzzy soft ring and t I m (f a ). Now x, y U (f a ; t), we have f a (x) t, f a (y) t. (FSR) f a (x + y) T { f a (x), f a (y)} T {t, t} t So, which implies x + y U (f a ; t). (FSR2) f a ( x) f a (x) t Therefore x U (f a ; t). f a (xy) T { f a (x), f a (y)} T {t, t} t Therefore xy U (f a ; t).

6 46 M. Marudai and V. Rajendran Hence, U (f a ; t) is a fuzzy soft ring of R. Proposition 3.5 If s a fuzzy soft ring of R and defined by f a + (x) = f a (x) + f a (0), for all x R, then f a + (x) is normal fuzzy soft ring of R which contains f a. For any x, y R. It follows that f + a (x) = f a (x) + f a (0) (FSR) f + a (x + y) = f a (x + y) + f a (0) T {{ f a (x), f a (y)}+ f a (0)} T {{ f a (x) + f a (0), f a (y) + f a (0),} T { f a + (x), f a + (y)} (FSR2) f a + ( x ) = f a ( x) + f a (0) f a (x) + f a (0) f a + (x) f a + (x y) = f a (x y) + f a (0) T {{ f a (x), f a (y)}+ f a (0)} T {{ f a (x) + f a (0), f a (y) + f a (0)} T { f a + (x), f a + (y)} Therefore, f a + is normal fuzzy soft ring of R. Proposition 3.6 If { f ai } i f a is a family of fuzzy soft rings of R, then whose = {(x, Λ (x) / x R}, where i f a. Let x, y R, then for i f a It follows that (FSR) ( x + y) = Λ (x + y) Λ T { f T {Λ T { a i (x), (x), Λ (x), (y)} (y)} (y)} fa i is fuzzy soft ring of R, (FSR2) ( x ) = Λ a i f ( x )

7 Fuzzy Soft Rings on Fuzzy Lattices 47 Λ (x) (x) ( xy ) = Λ a i f (xy ) Therefore, Λ T { f T { Λ T { a i (x), (x), Λ (x), (y)} (y)} (y)} fa i is fuzzy soft ring of R. Preposition 3.7 If s a fuzzy soft ring of R then Let x, y R. then (FSR) f c a (x + y) = f a (x + y) T {f a (x), f a (y)} S { f a (x), f a (y)} S { f a c (x), f c a (y)} f c a is a fuzzy soft ring of R. (FSR2) f c a ( x) = f a ( x) f a (x) f c a (x) f c a (xy) = f a (xy) T {f a (x), f a (y)} S { f a (x), f a (y)} Therefore, S { f a c (x), f c a (y)} f c a is fuzzy soft ring of R. Preposition 3.8 Let f a and f b be two fuzzy soft ring of R then f a U f b is fuzzy soft ring of R.

8 48 M. Marudai and V. Rajendran Let x, y R, then (FSR) (f a U f b ) (x + y ) = S { f a (x + y), f b (x + y)} S { T { f a (x), f b (y)}, T {{ f b (x), f b (y)}} T { S { f a (x), f a (y), f b (x), f b (y)}} T { S { f a (x), f b (x)}, S {f a (y), f b (y)}} T { (f a U f b ) (x), (f a U f b ) (y)} (FSR2) (f a U f b ) ( x ) = S { f a ( x), f b ( x)} S { f a (x), f b (x)} ( f a U f b ) (x) (f a U f b ) (xy) = S { f a (xy), f b (xy)} S { T { f a (x), f a (y)}, T { f b (x), f b (y)}} T {S ( f a (x), f a (y), f b (x), f b (y)} T { S { f a (x), f b (x)}, S {f a (y), f b (y)}} T { (f a U f b ) (x), (f a U f b ) (y)} Therefore, f a U f b is fuzzy soft ring of R. Preposition 3.9 Let f a and f b be two fuzzy soft rings of R then f a f b is fuzzy soft ring of R. Let x, y R. then (FSR) (f a f b ) (x + y ) = T { f a (x + y), f b (x + y)} T { T { f a (x), f a (y)}, T {{ f b (x), f b (y)}} T { T { f a (x), f a (y), f b (x), f b (y)}} T { T { f a (x), f b (x)}, T {f a (y), f b (y)}} T { (f a f b ) (x), (f a f b ) (y)} (FSR2) (f a f b ) ( x ) = T { f a ( x), f b ( x)} T { f a (x), f b (x)} (f a f b ) (x) (f a f b ) (xy ) = T { f a (x y), f b (x y)}

9 Fuzzy Soft Rings on Fuzzy Lattices 49 T { T { f a (x), f a (y)}, T {{ f b (x), f b (y)}} T { T { f a (x), f a (y), f b (x), f b (y)}} T { T { f a (x), f b (x)}, T {f a (y), f b (y)}} T { (f a f b ) (x), (f a f b ) (y)} Therefore, f a f b is fuzzy soft ring of R. Preposition 3.0 If f,,, be fuzzy soft ring of the rings R, R 2,, R n respectively then f f f is fuzzy soft ring of R R 2 R n. a 2 a n Let X = (x, x 2,, x n ), Y = (y, y 2,, y n ) R (FSR) ( f = ( f = ( f = T { f (x + y ), T { T { f (x ), T { f (x n ), a n f (y ), (y n )}} T { T { f (x ), (y 2 ), T { T { f (x ), T{ f (y ), T { ( f ( f (y 2 ), ) (X + Y) ) ((x, x 2,, x n ) + (y, y 2,, y n )) ) (x + y, x 2 + y 2,, x n + y n ) (x 2 + y 2 ),, f (y )}, T { a 2 (x 2 ), (y n )}} (x 2 ), FSR satisfied in R R 2 R n. (FSR2) ( f ) ( X) = ( f ) ( (x, x 2,, x n )) (y n )}} f (x 2 ), (x n ), (x n )}, (x n + y n )} ) (x, x 2,, x n ), ) (y, y 2,, y n )} (y 2 )},,

10 50 M. Marudai and V. Rajendran = ( f ( x, x 2,, x n ) = T { f ( x ), T { f (x ), ( f ( f ( f = ( f ) ( x 2 ),, (x 2 ),, ( x n )} (x n )} ) (x, x 2,, x n ) ) (X) ) (XY) ) ((x, x 2,, x n ) (y, y 2,, y n )) = ( f ) (x y, x 2 y 2,, x n y n ) T { f (x y ), (x n y n )} T { T { f (x ), T { f (x n ), a n f (y ), (x 2 y 2 ),, (y n )}} T { T { f (x ), (y 2 ), T { T { f (x ), T{ f (y ), T { ( f ( f (y 2 ), T { ( f ( f Therefore, ( f f (y )}, T { a 2 (x 2 ), (y n )}} (x 2 ), (y n )}} f (x 2 ), (x n ), (x n )}, ) (x, x 2,, x n ), ) (y, y 2,, y n )} ) (Y)} ) (X), (y 2 )},, fa n is fuzzy soft ring of R R 2 R n. Preposition 3. Let R and R be two ring and θ:r R be a soft homomorphism. If f b is a fuzzy soft

11 Fuzzy Soft Rings on Fuzzy Lattices 5 ring of R then the pre-image θ (f b ) is fuzzy soft ring of R. Assume that f b is fuzzy soft ring of R. Let x, y R. (FSR) μ (x + y) = μ θ (f b ) f θ (x + y) b = μ f b ( θx + θ y) T { μ f b (θx), μ f b (θy)} T { μ (x), μ (y)} θ (f b ) θ (f b ) (FSR2) μ ( x) = μ θ (f b ) f θ ( x) b μf b θ(x) θ (f b ) μ (x) μ (x y) = μ θ (f b ) f b θ (x y) = μ f b ((θx) (θy)) T { μ f b θ(x), μf b θ(y)} T { μ (x), μ (y)} θ (f b ) θ (f b ) Therefore θ (f b ) is fuzzy soft ring of R. Preposition 3.2 Let θ: R R be an epimorphism and f b be fuzzy soft set in R. If θ (f b ) is fuzzy soft ring of R then f b is fuzzy soft ring of R. Proof Let x, y R. Then there exist a, b R such that θ (a) = x, θ (b) = y. It follows that. (FSR) μ θ(f b ) (x + y) = μ f b θ (x + y) = μ ( θx + θ y) f b T { μ T { f b (θx), μ f b μ θ(f b ) (x), (θy)} μ θ(f b ) (y)}

12 52 M. Marudai and V. Rajendran (FSR2) μ θ(f b ) ( x) = μf b μf b θ(x) μ θ(f b ) (x) θ ( x) μ θ(f b ) (x y) = μ f b θ (x y) = μ ((θx) (θy)) f b T { μ f b (θx), μ f b (θy)} T { μf b θ(x), μ f θ(y)} b T { μ θ(f b ) (x), μ θ(f b ) (y)} Therefore θ(f b ) is fuzzy soft ring of R. Proposition 3.3 If s fuzzy soft ring of R and θ:r R be a soft homomorphism of R then the fuzzy soft set Proof Let x, y R. f θ a = {(x, f θ a (x)) / x R} is fuzzy soft ring of R. (FSR) (FSR2) f θ a (x + y) = f a θ (x + y) = f a (θx + θy) T {f a (θx), f a (θy)} T { f θ a (x), f θ a (y)} f θ a ( x) = f a (θ ( x)) f a (θ(x)) f θ a (x) f θ a (x y) = f a (θ (x y)) = f a ((θx) (θy)) T {f a (θx), f a (θy)} T { f θ a (x), f θ a (y)} Therefore, f θ a is fuzzy soft ring of R. Proposition 3.4 Let f a be a fuzzy soft set over L, then s fuzzy soft ring over L if and only if for all a

13 Fuzzy Soft Rings on Fuzzy Lattices 53 A and for arbitrary α [0, ] with (f a ) α 0, then α level soft set (f a ) α is fuzzy soft ring over L. Proof Let f a be fuzzy soft ring over L. Then for each a A, s a fuzzy sub ring of L. For arbitrary α [0, ], (f a ) α 0. Let x, y (f a ) α. Then f a (x) α and f a (y) α. (FSR) (f a ) α (x + y) T {(f a ) α (x), (f a ) α (y)} T {f a (x), f a (y) } T {α, α} α Therefore, x + y (f a ) α (FSR2) (f a ) α ( x) {(f a ) α (x) (f a ) α (x) α Therefore, x (f a ) α (f a ) α (x y) T {(f a ) α (x), (f a ) α (y)} T {f a (x), f a (y) } T {α, α} Therefore, xy (f a ) α Therefore (f a ) α is a fuzzy soft ring of R. Proposition 3.5 Every imaginable fuzzy soft ring μ of R is fuzzy soft ring of R. Proof Assume that μ is imaginable fuzzy soft ring of R. Then we have μ (x + y) T { μ (x), μ (y)} and μ ( x) μ (x), μ (xy) T{ μ (x), μ (y)} for all x, y R. and so Since μ is imaginable, we have. min { μ (x), μ (y)}= T { min { μ (x), μ (y)}, min { μ (x), μ (y)}} T { μ (x), μ (y)} min { μ (x), μ (y)}

14 54 M. Marudai and V. Rajendran T { μ (x), μ (y)} = min { μ (x), μ (y)} It follows that μ (x + y) T{ μ (x), μ (y)} for all x, y R. Hence μ is fuzzy soft ring of R. Proposition 3.6 If μ is fuzzy soft ring of R and θ is endomorphism of R then R. Proof For any x, y R, we have (FSR) μ (x + y)) = μ (θ (x + y)) = μ (θx + θy) T {μ(θx), μ (θy)} T { μ (x), μ (y)} μ is fuzzy soft ring of (FSR2) μ ( x) = μ (θ ( x ) μ θ(x) μ (x) μ (xy) = μ θ (x y) = μ ((θx) (θy)) T {μ(θx), μ (θy)} T {μθ(x), μ θ(y)} T { μ (x), μ (y)} Therefore, μ is fuzzy soft ring of R. Proposition 3.7 Let T be a continuous t-norms and let f be a soft homomorphism on R. If μ is fuzzy soft ring of R then μ f is fuzzy soft ring of f(r). Let A = f (y ) and A 2 = A 2 = Consider the set. f (y 2 ) and f (y + y 2 ). Where y, y 2 f(r).

15 Fuzzy Soft Rings on Fuzzy Lattices 55 A + A 2 = {x R / x = + a 2 }, for some A and a 2 A 2 If x A + A 2 then x = x + x 2 for some x A and x 2 A 2. So that we have f(x) = f(x ) + f(x 2 ) = y + y 2 Therefore, x f (y + y 2 ) = A 2 Thus A + A 2 A 2 It follows that (FSR) μ f (y + y 2 ) = sup {μ (x) / x = sup {μ (x) / x A 2 } = sup {μ (x) / x A + A 2 } f (y + y 2 )} sup {μ (x + x 2 ) / x A, x 2 A 2 } Since T is continuous, therefore for every > 0, we see that if Sup {{μ (x ) / x A } + x * δ and Sup {{μ (x 2 ) / x 2 A 2 } + x 2 * δ T { Sup {{μ (x ) / x A }, Sup {{μ (x 2 ) / x 2 A 2 }} + T (x *, x 2 *) Choose A and a 2 A 2 such that Sup {{μ (x ) / x A } + μ ( ) δ and Sup {{μ (x 2 ) / x 2 A 2 } + μ (a 2 ) δ Τ { Sup {{μ (x ) / x A }, Sup {{μ (x 2 ) / x 2 A 2 }} + T (μ ( ), μ (a 2 )) Consequently, we have μ f (y + y 2 ) sup {T {μ (x ), μ (x 2 ) / x A, x 2 A 2 }} Τ {sup {μ (x )/ x A }, sup { μ (x 2 ) / x 2 A 2 }} Τ {μ f (y ), μ f (y 2 ) } Similarly, we can show that, μ f ( x) μ f (x), and μ f (xy) T{μ f (x), μ f (y)} Hence μ f is fuzzy soft ring of f(r). Proposition 3.8 Onto homomorphic image of fuzzy soft ring with sup-property is fuzzy soft ring of R. Proof Let f: R R be an onto homomorphism of rings and μ be a sup property of fuzzy soft ring of R. Let x, y R and x 0 f (x ) and y 0 f (y ) be such that

16 56 M. Marudai and V. Rajendran μ (x 0 ) = sup μ (h) and μ (y 0 ) = sup μ (h) h f (x ) h f (y ) respectively, then we can deduce that (FSR) μ f (x + y ) = sup μ (z) z f (x + y ) T {μ(x 0 ), μ(y 0 )} = T { sup μ (h), sup μ (h) } h f (x ) h f (y ) T {μ f ( x ), μ f (y )} (FSR2) μ f ( x ) = sup μ (z) z f ( x ) μ(x 0 ) = sup μ (h) h f (x ) μ f ( x ) μ f (x y ) = sup μ (z) z f (x y ) T {μ(x 0 ), μ(y 0 )} = T { sup μ (h), sup μ (h) } h f (x ) h f (y ) T {μ f ( x ), μ f (y )} Therefore μ f is fuzzy soft ring of R. Proposition 3.9 Let f a be a fuzzy soft ring over R and (φ, Ψ) be a fuzzy soft homomorphism from R to R. Then (φ, Ψ)s fuzzy soft ring over R. Let k (Ψ)f a and y, y 2 Y. If φ (y ) = φ or φ (y 2 ) = φ. The proof is straight forward. Let us assume that, there exist x, x 2 X such that φ (x ) = y and φ (x 2 ) = y 2.

17 Fuzzy Soft Rings on Fuzzy Lattices 57 Now φ (f a ) k (y + y 2 ) = V V f a (t) φ (t) = y + y 2 Ψ (a) = k V f a (x + x 2 ) Ψ (a) = k V {T {f a (x ), f a (x 2 )} Ψ (a) = k T { V {f a (x ), V f a (x 2 )} Ψ (a) = k Ψ (a) = k This inequality is satisfy for each x, x 2 X, which satisfy φ (x ) = y, φ (x 2 ) = y 2, then we have (FSR) φ (f a ) k (y + y 2 ) T { ( V, V φ (t ) = y Ψ (a) = k ( V, V (t 2 ) ) } φ (t 2 ) = y 2 Ψ (a) = k f (t ) ), T {φ (f a ) k (y ), φ (f a ) k (y 2 )} And similarly, we can have φ (f a ) k ( y) φ (f a ) k (y), φ (f a ) k (y y 2 ) T {φ (f a ) k (y ), φ (f a ) k (y 2 )} Therefore (φ, Ψ)s fuzzy soft ring of R. Proposition 3.20 Let g b be a fuzzy soft ring over R and (φ, Ψ) be a fuzzy soft homomorphism from R to R, then ( φψ, ) g b is fuzzy soft ring over R. Proof Let a Ψ (B) and x, x 2 X. (FSR) φ (g a ) (x + x 2 ) = g Ψ (a) (φ (x. x 2 ) = g Ψ (a) (φ (x ). (x 2 )) T { g Ψ (a) φ(x ). g Ψ (a) φ (x 2 )}

18 58 M. Marudai and V. Rajendran T {φ (g a ) (x ), φ (g a ) (x 2 )} and similarly, we have φ (g a ) ( x) φ (g a ) (x) and φ (g a ) (x x 2 ) {φ (g a ) (x ), φ (g a ) (x 2 )} Therefore, ( φψ, ) g b is fuzzy soft ring over R. Conclusions This paper summarized the basic concept of fuzzy soft set. We then presented a detailed theoretical study of fuzzy soft set, which led to the definition of new algebraic structures in ring structures. This work focused on fuzzy soft rings, homomorphism of fuzzy soft rings, and pre-image of fuzzy soft rings. To extend this work one could study the properties of fuzzy soft sets in other algebraic structures such as near rings, Groups, ideals, fields and G-modules. References [] U.Acar, F. Koyuncu and B. Tanay, Soft sets andsoft rings, Computers and Mathematics with Applications, 59(200) [2] B Ahmat and A Kharal, On fuzzy soft sets, Hindawi Publishing Corporation Advances in Fuzzy Systems (2009), Article ID , 6 pages. [3] H. Aktas and N.Cagman, Soft sets and soft group, Information Science, 77 (2007) [4] M.I. Ali, F. Feng, X Liu and W.K.M. Shabir, On some new operations in soft set theory, Computers and Mathematics with Applications 57 (2009) [5] K. Atanassov, Intuitionistic fuzzy Sets and Systems 64(2) (986) [6] A. Aygunoglu and H. Aygun, Introduction to fuzzy soft groups, Computers and Mathematics with Applications 58 (2009) [7] A. Aygunoglu and H.Aygun, Soft sets and soft topological spaces, Submitted. [8] K.V. Babitha and J.J. Sunil, Soft set relations and functions, Computers and Mathematics with Applications 60 (200) [9] N.Cagman and S. Enginoglu, Soft matrix theory and its decision making, Computers and Mathematics with Applications 59(200) [0] D. Chen, E.C.C. Tsang, D.S. Yeung and X. Wang, The parameterization reduction of soft set and its applications, Computers and Mathematics with applications 49 (2005) [] V.N. Dixit,R. Kumar and N. Ajmal, on fuzzy rings, Fuzzy Sets and systems 49 (992) [2] F.Feng, Y.B. Jun and X. Zhao, Soft semirings, Computers and Mathematics with Applications 56 (2008)

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