International Journal of Mathematical Archive-5(8), 2014, Available online through ISSN

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1 nternational Journal of Mathematical Archive-5(8), 2014, Available online through SSN A STUDY ON NTERVAL VALUED FUZZY SUBRNGS OF A RNG 1 M. G. Somasundara Moorthy*, 2 K. Arjunan and 3 R. Uthaya Kumar 1Department of Mathematics, Mookambigai College of Engineering, Keeranur, (T.N.), ndia. 2Department of Mathematics, H.H.The Rajahs College, Pudukkottai , (T.N.), ndia. 3Professor of Mathematics, Gandhigram Rural University, Gandhigram , (T.N.), ndia. (Received On: ; Revised & Accepted On: ) ABSTRACT n this paper, we study some of the properties of interval valued fuzzy subring of a ring and prove some results on these AMS SUBJECT CLASSFCATON: 03F55, 08A72, 20N25. Key Words: nterval valued fuzzy subset, interval valued fuzzy subring, interval valued fuzzy relation, Product of interval valued fuzzy subsets. NTRODUCTON nterval-valued fuzzy sets were introduced independently by Zadeh [12], Grattan-Guiness [5], Jahn [7], in the seventies, in the same year. An interval valued fuzzy set (VF) is defined by an interval-valued membership function. Jun.Y.B and Kin.K.H [8] defined an interval valued fuzzy R-subgroups of nearrings. Solairaju.A and Nagarajan.R [10] defined the charactarization of interval valued Anti fuzzy Left h-ideals over Hemirings. Azriel Rosenfeld [3] defined a fuzzy group. Asok Kumer Ray [2] defined a product of fuzzy subgroups.we introduce the concept of interval valued fuzzy subring of a ring and established some results. 1. PRELMNARES Definition 1.1: Let X be any nonempty set. A mapping [M]: X D[0, 1] is called an interval valued fuzzy subset (briefly, VFS ) of X, where D[0,1] denotes the family of all closed subintervals of [0,1] and [M](x) = [M (x), M + (x)], for all x in X, where M and M + are fuzzy subsets of X such that M (x) M + (x), for all x in X. Thus [M] (x) is an interval (a closed subset of [0, 1] ) and not a number from the interval [0,1] as in the case of fuzzy subset. Note that [0] = [0, 0] and [1] = [1, 1]. Remark 1.2: Let D X be the set of all interval valued fuzzy subset of X, where D means D[0, 1]. Definition 1.3: Let [M] = { x, [M (x), M + (x)] / x X}, [N] = { x, [N (x), N + (x)] / x X} be any two interval valued fuzzy subsets of X. We define the following relations and operations: (i) [M] [N] if and only if M (x) N (x) and M + (x) N + (x), for all x in X. (ii) [M] = [N] if and only if M (x) = N (x) and M + (x) = N + (x), for all x in X. (iii) [M] [N] = { x, [ min { M (x), N (x)}, min { M + (x), N + (x)} ] / x X }. (iv) [M] [N] = { x, [ max { M (x), N (x)}, max { M + (x), N + (x)} ] / x X }. (v) [M] C = [1] [M] = { x, [ 1 M + (x), 1- M (x)] / x X }. Definition 1.4: Let (R, +, ) be a ring. An interval valued fuzzy subset [M] of R is said to be an interval valued fuzzy subring (VFSR) of R if the following conditions are satisfied: (i) [M](x+y) rmin { [M](x), [M](y) }, (ii) [M]( x) [M](x), (iii) [M](xy) rmin { [M](x), [M](y) }, for all x and y in R. Corresponding author: 1 M. G. Somasundara Moorthy* nternational Journal of Mathematical Archive- 5(8), August

2 1 M. G. Somasundara Moorthy*, 2 K. Arjunan and 3 R. Uthaya Kumar/A Study On nterval Valued Fuzzy Subrings Of A Ring / JMA- 5(8), August Definition 1.5: Let [M] and [N] be any two interval valued fuzzy subsets of sets R and H, respectively. The product of [M] and [N], denoted by [M] [N], is defined as[m] [N] ={ (x, y), [M] [N](x, y) / for all x in R and y in H}, where [M] [N](x, y) = rmin { [M](x), [N](y) }. Definition 1.6: Let [M] be an interval valued fuzzy subset in a set S, the strongest interval valued fuzzy relation on S, that is an interval valued fuzzy relation [V] with respect to [M] given by [V](x, y) = rmin { [M](x), [M](y) }, for all x and y in S. 2. PROPERTES OF NTERVAL VALUED FUZZY SUBRNGS Theorem 2.1: f [M] is an interval valued fuzzy subring of a ring (R, +, ), then [M]( x) = [M](x) and [M](x) [M](e), for x in R, the identity element e in R. Proof: For x in R and e is the identity element of R. Now, [M] (x) = [M] ( ( x)) [M]( x) [M](x). Therefore, [M] ( x) = [M] (x), for all x in R. Now, [M] (e) = [M] (x x) rmin {[M](x), [M]( x)} = [M]( x). Therefore, [M](e) [M](x), for x in R. Theorem 2.2: f [M] is an interval valued fuzzy subring of a ring (R, +, ), then [M] (x y) = [M] (e) gives [M] (x) = [M] (y), for x and y in R, the identity element e in R. Proof: Let x and y in R, the identity element e in R. Now, [M](x) = [M](x y+y) rmin {[M](x y), [M](y)} = rmin {[M](e), [M](y) } = [M](y) = [M](x (x y)) rmin {[M](x y), [M](x) } = rmin { [M](e), [M](x) } = [M](x). Therefore, [M](x) = [M](y), for x and y in R. Theorem 2.3: f [M] is an interval valued fuzzy subring of a ring (R, +, ), then H = {x / x R: [M](x) = [1] } is either empty or is a Proof: f no element satisfies this condition, then H is empty. f x and y in H, then [M](x y) rmin {[M](x), [M](y)} = rmin {[1], [1]} = [1]. Therefore, [M] (x y) = [1]. We get x y in H. And [M] (xy) rmin {[M](x), [M](y)} = rmin {[1], [1]} = [1]. Therefore, [M] (xy) = [1]. We get xy in H. Therefore, H is a Hence H is either empty or is a Theorem 2.4: f [M] is an interval valued fuzzy subring of a ring (R, +, ), then H = {x R: [M] (x) = [M] (e)} is a Proof: Let x and y be in H. Now, [M](x y) rmin {[M](x), [M](y) } = rmin { [M](e), [M](e)} = [M](e). Therefore, [M](x y) [M](e). Hence [M](e) = [M](x y ). Therefore, x y in H. And, [M](xy) rmin { [M](x), [M](y) } = rmin { [M](e), [M](e) } = [M](e). Therefore, [M](xy) [M](e). Hence [M](e) = [M](xy). Therefore, xy in H. Hence H is a Theorem 2.5: Let [M} be an interval valued fuzzy subring of a ring (R, +, ). f [M](x y) = [1], then [M](x) = [M](y), for x and y in R. Proof: Let x and y in R. Now, [M](x) = [M](x y+y) rmin { [M](x y), [M](y)} = rmin{[1], [M](y)} = [M](y) = [M]( y) = [M]( x+x y) rmin {[M](x), [M](x y)} = rmin {[M]( x), [1] } = [M](x). Therefore, [M] (x) = [M] (y), for x and y in R. Theorem 2.6: Let [M] be an interval valued fuzzy subring of a ring (R, +, ). f [M](x y) = [0], then either [M](x) = [0] or [M](y) = [0], for x and y in R. Proof: Let x and y in R. By the definition [M](x y) rmin {[M](x), M](y)}, which implies that [0] rmin {[M](x), [M](y)}. Therefore, either [M](x) = [0] or [M](y) = [0]. Theorem 2.7: f [M] and [N] are two interval valued fuzzy subrings of a ring R, then their intersection [M] [N] is an interval valued fuzzy Proof: Let x and y belong to R, [M] = { x, [M] (x) / x in R} and [N] = { x, [N](x) / x in R}. Let [K] = [M] [N] and [K]={ x, [K](x) / x in R}. (i) [K](x y) = rmin{[m](x y), [N](x y)} rmin{rmin{[m](x), [M](y)}, rmin{[n](x), [N](y) }} = rmin {rmin {[M](x), [N](x)}, rmin {[M](y), [N](y)}} = rmin { [K](x), [K](y)}. Therefore, [K](x y) rmin { [K](x), [K](y)}, for all x and y in R. (ii) [K](xy) = rmin {[M](xy), [N](xy)} rmin {rmin {[M](x), [M](y)}, rmin {[N](x), [N](y)}} = rmin {rmin {[M](x), [N](x)}, rmin {[M](y), [N](y)}} = rmin {[K](x), [K](y)}. 2014, JMA. All Rights Reserved 72

3 1 M. G. Somasundara Moorthy*, 2 K. Arjunan and 3 R. Uthaya Kumar/A Study On nterval Valued Fuzzy Subrings Of A Ring / JMA- 5(8), August Therefore, [K](xy) rmin {[K](x), [K](y) }, for all x and y in R. Hence [M] [N] is an interval valued fuzzy subring of the ring R. Theorem 2.8: The intersection of a family of interval valued fuzzy subrings of a ring R is an interval valued fuzzy Proof: Let {[ ]} be a family of interval valued fuzzy subrings of a ring R and [M] = [ ]. Then for x and y belongs to R, we have (i) [M](x y) = [ ]( x y) rmin {[ ](x), [ ](y) } rmin { rmin {[M](x), [M](y) }. Therefore, [M](x y) rmin {[M](x), [M](y) }, for all x and y in R. ]( xy) [ ]( x) ]( x), [ ]( y) ]( y) } = (ii) [M](xy) = rmin {[ ](x), [ ](y)} rmin{, {[M](x), [M](y)}. Therefore, [M](xy) rmin {[M](x), [M](y)}, for all x and y in R. Hence the intersection of a family of interval valued fuzzy subrings of the ring R is an interval valued fuzzy }= rmin Theorem 2.9: Let [M] be an interval valued fuzzy subring of a ring R. f [M](x) < [M](y), for some x and y in R, then [M](x+y) = [M](x) = [M](y+x), for all x and y in R. Proof: Let [M] be an interval valued fuzzy subring of a ring R. Also we have [M](x) < [M](y), for some x and y in R, [M](x+y) rmin {[M](x), [M](y)} = [M](x); and [M](x) = [M](x+y y) rmin {[M](x+y), [M](y)} = [M](x+y). Therefore, [M](x+y) = [M](x), for all x and y in R. Hence [M](x+y) = [M](x) = [M](y+x), for all x and y in R. Theorem 2.10: Let [M] be an interval valued fuzzy subring of a ring R. f [M](x) > [M](y), for some x and y in R, then [M](x+y) = [M](y) = [M](y+x), for all x and y in R. Proof: t is trivial. Theorem 2.11: Let [M] be an interval valued fuzzy subring of a ring R such that m [M] = {α}, where α in D[0, 1]. f [M] = [K] [L],where [K] and [L] are interval valued fuzzy subrings of R, then either [K] [L] or [L] [K]. Proof: Let [M] = [K] [L] = { x, [M](x) / x in R}, [K] = { x, [K](x) / x in R} and [L] = { x, [L](x) /x in R}. Suppose that neither [K] [L] nor [L] [K]. Assume that [K](x) > [L](x) and [K](y) < [L](y), for some x and y in R. Then, α = [M](x) = ([K] [L])(x) = rmax {[K](x), [L](x) }= [K](x) > [L](x). Therefore, α > [L](x). And, α = [M](y) = ([K] [L])(y) = rmax {[K](y), [L](y) }= [L](y) > [K](y). Therefore, α > [K](y). So that, [L](y) > [L](x) and [K](x) > [K](y). Hence [K](x+y) = [K](y) and [L](x+y) = [L](x), by Theorem 2.9 and But then, α = [M](x+y) = rmax { ([K] [L])(x+y) = rmax { [K](x+y), [L](x+y) }= rmax {[K](y), [L](x) }< α (1). t is a contradiction by (1). Therefore, either [K] [L] or [L] [K] is true. Theorem 2.12: f [M] and [N] are interval valued fuzzy subrings of the rings R and H, respectively, then [M] [N] is an interval valued fuzzy subring of R H. Proof: Let [M] and [N] be interval valued fuzzy subrings of the rings R and H respectively. Let x 1 and x 2 be in R, y 1 and y 2 be in H. Then (x 1, y 1 ) and (x 2, y 2 ) are in R H. Now, [M] [N] [(x 1, y 1 ) (x 2, y 2 )] = [M] [N](x 1 x 2, y 1 y 2 ) = rmin {[M](x 1 x 2 ), [N](y 1 y 2 ) } rmin { rmin { [M](x 1 ), [M](x 2 )}, rmin {[N](y 1 ), [N](y 2 )}} = rmin {rmin {[M](x 1 ), [N](y 1 )}, rmin {[M](x 2 ), [N](y 2 )}}= rmin{[m] [N](x 1, y 1 ), [M] [N](x 2, y 2 )}. Therefore, [M] [N][(x 1, y 1 ) (x 2, y 2 )] rmin {[M] [N](x 1, y 1 ), [M] [N](x 2, y 2 )}. and, [M] [N][(x 1, y 1 )(x 2, y 2 )]=[M] [N](x 1 x 2, y 1 y 2 ) = rmin {[M](x 1 x 2 ), [N](y 1 y 2 ) } rmin {rmin{[m](x 1 ), [M](x 2 )}, rmin{[n](y 1 ), [N](y 2 )}}=rmin{rmin{[m](x 1 ), [N](y 1 )}, rmin{[m](x 2 ), [N](y 2 )}}= rmin {[M] [N](x 1,y 1 ), [M] [N](x 2, y 2 )}. Therefore, [M] [N][(x 1, y 1 )(x 2, y 2 )] rmin{[m] [N](x 1, y 1 ), [M] [N](x 2, y 2 )}. Hence [M] [N] is an interval valued fuzzy subring of R H. Theorem 2.13: Let [M] and [N] be interval valued fuzzy subsets of the rings R and H, respectively. Suppose that e and e are the identity element of R and H, respectively. f [M] [N] is an interval valued fuzzy subring of R H, then at least one of the following two statements must hold. (i) [N](e ) [M](x), for all x in R, (ii) [M](e) [N](y), for all y in H. Proof: Let [M] [N] be an interval valued fuzzy subring of R H. 2014, JMA. All Rights Reserved 73

4 1 M. G. Somasundara Moorthy*, 2 K. Arjunan and 3 R. Uthaya Kumar/A Study On nterval Valued Fuzzy Subrings Of A Ring / JMA- 5(8), August By contra positive, suppose that none of the statements (i) and (ii) holds. Then we can find a in R and b in H such that [M](a) > [N](e ) and [N](b) > [M](e). We have, [M] [N](a, b) = rmin {[M](a), [N](b)} > rmin {[M](e), [N](e )} = [M] [N](e, e ). Thus [M] [N] is not an interval valued fuzzy subring of R H. Hence either [N](e ) [M](x), for all x in R or [M](e) [N](y), for all y in H. Theorem 2.14: Let [M] and [N] be interval valued fuzzy subsets of the rings R and H, respectively and [M] [N] is an interval valued fuzzy subring of R H. Then the following are true: (i) if [M](x) [N](e ), then [M] is an interval valued fuzzy (ii) if [N](x) [M](e), then [N] is an interval valued fuzzy subring of H. (iii) either [M] is an interval valued fuzzy subring of R or [N] is an interval valued fuzzy subring of H. Proof: Let [M] [N] be an interval valued fuzzy subring of R H and x, y be in R. Then (x, e ) and (y, e ) are in R H. Now, using the property [M](x) [N](e ), for all x in R, we get, [M](x y) = rmin{ [M](x y), [N](e e )} = [M] [N]((x y), (e e )) = [M] [N][(x, e ) +( y, e )] rmin {[M] [N](x, e ), [M] [N]( y, e ) } = rmin {rmin {[M](x), [N](e )}, rmin {[M]( y), [N](e )}} = rmin {[M](x), [M]( y) } rmin {[M](x), [M](y)}. Therefore, [M](x y) rmin { [M](x), [M](y) }, for all x, y in R. And, [M](xy) = rmin {[M](xy), [N](e e )} = [M] [N]( (xy), (e e )) = [M] [N][ (x, e )(y, e )] rmin {[M] [N](x, e ), [M] [N](y, e )} = rmin {rmin{[m](x), [N](e ) }, rmin {[M](y), [N](e )}} = rmin {[M](x), [M](y)} rmin {[M](x), [M](y)}. Therefore, [M](xy) rmin {[M](x), [M](y)}, for all x, y in R. Hence [M] is an interval valued fuzzy Thus (i) is proved. Now, using the property [N](x) [M](e), for all x in H, we get, [N](x y) = rmin {[N](x y), [M](ee)} = [M] [N]( (ee), (x y)) = [M] [N][ (e, x)+(e, y)] rmin {[M] [N](e, x), [M] [N](e, y)} = rmin {rmin {[N](x), [M](e)}, rmin {[N]( y), [M](e)}} = rmin {[N](x), [N]( y)} rmin {[N](x), [N](y)}. Therefore, [N](x y) rmin {[N](x), [N](y)}, for all x and y in H. And, [N](xy) = rmin {[N](xy), [M](ee)} = [M] [N]( (ee), (xy)) = [M] [N][(e, x)(e, y)] rmin {[M] [N](e, x), [M] [N](e, y)} = rmin {rmin {[N](x), [M](e)}, rmin { [N](y), [M](e)}} = rmin {[N](x), [N](y) } rmin {[N](x), [N](y)}. Therefore, [N](xy) rmin {[N](x), [N](y)}, for all x and y in H. Hence [N] is an interval valued fuzzy subring of H. Thus (ii) is proved. (iii) is clear. Theorem 2.15: Let [M] be an interval valued fuzzy subset of a ring R and [V] be the strongest interval valued fuzzy relation of R with respect to [M]. Then [M] is an interval valued fuzzy subring of R if and only if [V] is an interval valued fuzzy subring of R R. Proof: Suppose that [M] is an interval valued fuzzy Then for any x = (x 1, x 2 ) and y = (y 1, y 2 ) are in R R. We have, [V](x y) = [V][(x 1, x 2 ) (y 1, y 2 )] = [V](x 1 y 1, x 2 y 2 ) = rmin {[M](x 1 y 1 ), [M](x 2 y 2 ) } rmin {rmin {[M](x 1 ), [M]( y 1 )}, rmin {[M](x 2 ), [M]( y 2 )}} = rmin {rmin{[m](x 1 ), [M](x 2 ) }, rmin {[M]( y 1 ), [M]( y 2 ) }} = rmin {rmin {[M](x 1 ), [M](x 2 )}, rmin {[M](y 1 ), [M](y 2 )}} = rmin {[V](x 1, x 2 ), [V](y 1, y 2 )} = rmin {[V](x), [V](y)}. Therefore, [V](x y) rmin {[V](x), [V](y)}, for all x and y in R R. And we have, [V](xy) = [V][(x 1, x 2 )(y 1, y 2 )] = [V](x 1 y 1, x 2 y 2 ) = rmin { [M](x 1 y 1 ), [M](x 2 y 2 )} rmin {rmin {[M](x 1 ), [M](y 1 )}, rmin{[m](x 2 ), [M](y 2 )}}= rmin {rmin {[M](x 1 ), [M](x 2 )}, rmin {[M](y 1 ), [M](y 2 )}} = rmin {[V](x 1, x 2 ), [V](y 1, y 2 )} = rmin {[V](x), [V](y)}. Therefore, [V](xy) rmin {[V](x), [V](y)}, for all x and y in R R. This proves that [V] is an interval valued fuzzy subring of R R. Conversely, assume that [V] is an interval valued fuzzy subring of R R, then for any x = (x 1, x 2 ) and y = (y 1, y 2 ) are in R R, we have rmin {[M](x 1 y 1 ), [M](x 2 y 2 )} = [V](x 1 y 1, x 2 y 2 ) = [V][(x 1, x 2 ) (y 1, y 2 )] = [V](x y) rmin {[V](x), [V](y)}= rmin {[V](x 1, x 2 ), [V](y 1, y 2 ) } = rmin { rmin {[M](x 1 ), [M](x 2 )}, rmin {[M](y 1 ), [M](y 2 )}}. f we put x 2 = y 2 = e, where e is the identity element of R. We get, [M](x 1 y 1 ) rmin { [M](x 1 ), [M](y 1 ) }, for all x 1 and y 1 in R. And rmin { [M](x 1 y 1 ), [M](x 2 y 2 )} = [V](x 1 y 1, x 2 y 2 ) = [V][(x 1, x 2 )(y 1, y 2 )] = [V](xy) rmin {[V](x), [V](y)} = rmin {[V](x 1, x 2 ), [V](y 1, y 2 )} = rmin {rmin {[M](x 1 ), [M](x 2 )}, rmin {[M](y 1 ), [M](y 2 )}}. f we put x 2 = y 2 = e, where e is the identity element of R. We get, [M](x 1 y 1 ) rmin {[M](x 1 ), [M](y 1 ) }, for all x 1 and y 1 in R. Hence [M] is an interval valued fuzzy REFERENCES 1. 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