Homomorphism on T Anti-Fuzzy Ideals of Ring

Transcription

1 International Journal o Computational Science and Mathematics. ISSN Volume 8, Number 1 (2016), pp International esearch Publication House Homomorphism on T nti-fuzzy Ideals o ing 1 J. Prakashmanimaran, 2 B.Chellappa and 3 M. Jeyakumar `1 esearch Scholar (Part Time - Mathematics), Manonmaniam Sundharanar University, Tirunelveli, Tamilnadu, India 2 Principal, Nachiappa Suvamical rts and Science College, Karaikudi , Tamilnadu, India 3 ssistant Proessor, Dept. o Mathematics, lagappa University Evening College, ameswaram , Tamilnadu, India bstract In this paper, we made an attempt to study the properties o T uzzy ideal o a ring and we introduce some theorems an homomorphic image, an epimorphic pre-image o a T anti-uzzy ideal o a ring. Keywords: Fuzzy subset, T uzzy ideal, homomorphism o T uzzy ideal, homomorphism o T anti-uzzy ideal, homomorphic image o T anti-uzzy ideal, homomorphic pre-image T anti-uzzy ideal. INTODUCTION In 1965, Zadeh introduced the concept o uzzy sets. In 1967, oseneld deined the idea o uzzy subgroups and gave some o its properties. Biswas introduced the notion o anti uzzy subgroups. In 1982, Wang-jin Liu introduced the concept o uzzy ring and uzzy ideal and discussed the operations on uzzy ideals. Fuzzy subnear-rings are introduced by bou-zaid. In W.. Dudek and Y. B. Jun introduced uzzy subgroups over a t norm. Many researchers are engaged in extending the concepts. Chandrasekhara ao. K and V. Swaminathan [3] deined the anti-homomorphisms in

2 36 J. Prakashmanimaran, B.Chellappa and M. Jeyakumar near rings. In the year 1998, Sung M.H. et al. [12] proved the same result using the level uzzy subsets and obtained some properties based on near-ring homomorphism. Homomorphic images and pre images o anti uzzy ideals are investigated by K.H. Kim et al. [9]. In this paper we deine, homomorphism and study the ring homomorphism. We introduced homomorphism in T anti-uzzy ideals o ring. We discuss some o its properties. We have shown that homomorphism, homomorphic image o T anti-uzzy ideal, homomorphic pre-image T anti-uzzy ideal. Deinition: 1 nonempty set together with two binary operation and. is called a ring i (i)., is an abelian group, (ii).,. is a semigroup (iii). ; x y z xy xz x y z xz yz, or all x, y, z in Deinition: 2 non-empty set is called lattice ordered ring or operations,,, deined on it and satisy the ollowing Example: 1 (i),, is a ring (ii),, is a lattice ring i it has our binary (iii) x y z x y x z; x y z x y x z y z x y x z x; y z x y x z x (iv) x y z x y x z ; x y z x y x z y z x y x z x ; y z x y x z x or all x, y, z in and x 0, Z,,,, is a ring, where Z is the set o all integers.

3 Homomorphism on T nti-fuzzy Ideals O Near-ing 37 Example: 2 nz,,,, is a ring, where Z is the set o all integers and n Z Deinition: 3 mapping T : 0, 1 0, 1 0, 1 is called a triangular norm [ t norm] i and only i it satisies the ollowing conditions: Deinition: 4 (i). T x, 1 T 1, x x, or all x 0, 1 (ii). i x x *, y y * then T x, y T x *, y * (iii). T x y T y x x (iv).,,, or all, y 0, 1. T x, T y, z T T x, y, z. n anti- uzzy subset o a ring is called T anti-uzzy right (resp. let) ideal i (i), max, x y T x y x y (ii) x y x (resp. let x y y ), or all x, y in Deinition: 5 n anti- uzzy subset ring) is called an antiuzzy sub o a lattice ordered ring (or ring o, i the ollowing conditions are satisied (i) x y max x, y (ii) x y max x, y (iii) x y max x, y (iv) x y max x, y, or all x, y in Deinition: 6 n anti-uzzy subset o a ollowing conditions are satisied, ring is called a T anti-uzzy ideal, i the

4 38 J. Prakashmanimaran, B.Chellappa and M. Jeyakumar (i) x y T x, y (ii) x y x; x y y (iii) x y T x, y (iv) x y T x, y, or all x, y Example: 3 Now a b c,,,,,, is a ring. The operations,, and deined by the ollowing tables Consider the anti-uzzy subset o the Then is a T anti-uzzy ideal o ring 0.3 i x a x 0.5 i x b 0.7 i x c ring Deinition: 7 Let 1 and 2 be two rings. Then the unction : 1 2 is called a ring homomorphism i satisies the ollowing conditions (i) x y x y (ii) x y x y (iii) x y x y (iv) x y x y, or all x, y in Example: 4 Let m n 2 or all m, n Z is a ring under usual addition and multiplication Deine : by

5 Homomorphism on T nti-fuzzy Ideals O Near-ing 39 m n 2 m n 2 is ring homomorphism, where Z is set o all integer Deinition: 8 Let 1 and 2 be two rings. mapping : 1 2 is called a ring isomorphism i (i) is one-to-one (ii) is onto, or all x, y in Deinition: 9 n anti-uzzy set there exists a a0 o a ring has the in property i or any subset N o, N such that a0 in a an Deinition: 10 Let M and N be any two sets and let : M N be any unction. n anti-uzzy subset o a M is called -invariant i x yimplies x y or all x, y M, Deinition: 11 Let be a ring. Let be an anti-uzzy set o a T anti-uzzy ideals o a ring and be a unction deined on, then the anti-uzzy set deined by under y in x, or all y 1 n y in is and is called the image o Deinition: 12 Let be a ring. Let be an anti-uzzy set o a T anti-uzzy ideals o a ring and be a unction deined on, i v is a uzzy set in, then

6 40 J. Prakashmanimaran, B.Chellappa and M. Jeyakumar in is deined by v image o v under x v x, or all x and is called the pre- Theorem: 1 n onto homomorphism image o a T anti-uzzy ideal o a property is a T anti-uzzy ideal o a ring. Proo: Let and S be a rings Let : S be an epimorphism and be a T anti-uzzy ideal o a with in property. Let x, y S Let 0 1, 1 y y and z 1 z x x 1 n x 0 n x 0 in, 1 n y n y 0 in and 1 n z n z 0 in respectively, then We have (i) x y in z z 1 xy x y 0 0 max, x0 y0 0, 0 T x y 0 be such that T in n, in n n 1 x n 1 y T x, y Thereore x y T x, y, or all x, y S ring with in ring

7 Homomorphism on T nti-fuzzy Ideals O Near-ing 41 (ii) Let be a T uzzy ideals o and let x, y Since xy x and xy y lso we have xy in 1 z xy 0 0 x y x 0 in 1 n x x Thereore xy x (iii) n z, or all x, y S x y in z z 1 xy x y 0 0 max, x0 y0 0, 0 T x y T in n, in n n 1 x n 1 y T x, y Thereore x y T x, y (iv), or all x, y S x y in z z 1 xy x y 0 0 max, x0 y0 0, 0 T x y

8 42 J. Prakashmanimaran, B.Chellappa and M. Jeyakumar T in n, in n n 1 x n 1 y T x, y Thereore x y T x, y, or all x, y S Thus an onto homomorphic image o a T anti-uzzy ideal o a property is a T anti-uzzy ideal o a ring. ring with in Theorem: 2 n epimorphic pre-image o a T anti-uzzy ideal o a ideal o a ring. Proo: Let and S be a rings Let : S be an epimorphism ring is a T anti-uzzy Let v be a T anti-uzzy ideal o a ring S and be the pre-image o v under or any x, y, z. (i) we have x y v x y v x y v x y, T v x v y, T v x v y T x, y Thereore x y T x, y, or all x, y (ii) Since xy x and xy y xy v xy v xy

9 Homomorphism on T nti-fuzzy Ideals O Near-ing 43 v x y T v x T v x v x x Thereore xy x, or all x, y (iii) we have x y v x y v x y v x y, T v x v y, T v x v y T x, y Thereore x y T x, y, or all x, y (iv) we have x y v x y v x y v x y, T v x v y, S T v x v y T x, y Thereore x y T x, y, or all x, y Thus an epimorphic pre-image o a T anti-uzzy ideal o a ring is a T antiuzzy ideal o a ring.

10 44 J. Prakashmanimaran, B.Chellappa and M. Jeyakumar Proposition: 1 Let and S be rings and let : S be a homomorphism. Let be invariant anti-uzzy ideal o a ring. I x a, then ax a or all a. Theorem: 3, Let : S be an epimorphism o rings and S. I is invariant antiuzzy ideal o a ring, then is a T anti-uzzy ideal o S. Proo: Let a, b, c S then there exists x, y, z such that x a, y b and Suppose is invariant anti-uzzy ideal o a ring, then by Proposition 1 (i) we have a b x y x y x y T x, y, T a b Thereore a b T a, b x, y z c, or all a, bs and (ii) Since xy x and xy y We have ab x y xy xy x a

11 Homomorphism on T nti-fuzzy Ideals O Near-ing 45 Thereore ab a, or all a, b S, x, y (iii) we have a b x y x y x y T x, y, T a b Thereore a b T a, b x, y, or all a, bs and (iv) we have a b x y x y x y T x, y, T a b Thereore a b T a, b x, y, or all a, bs and Hence is a T anti-uzzy ideal o a ring S. Theorem: 4 Let : 1 2 be an onto homomorphism o a rings. I is T anti-uzzy ideal o 1 Proo:, then is a T anti-uzzy ideal o 2 Let be a T anti-uzzy ideal o a ring Let 1 y1 and 2 y2 Similarly 1 3 y1 y 2., where y1, y 2 2 are non-empty subsets o 2

12 46 J. Prakashmanimaran, B.Chellappa and M. Jeyakumar Consider the set 1 2 a1 a2 / a, a2 2 I x1 2, then x x1 x2, or some x, x2 2 and so, 1 2 x x x 1 2 x x y y x y y Thus that is x / x y1y 2 x1 x2 / x1 y1, x2 y 2 Let y3 2, then 1 (i) we have y1 y 2 in x / x y1 y in x x / x y, x y x1 x 2 x1 y1 x 2 y 2 in max, /, in T x, x / x y, x y T in x / x y, in x / x y 1, 2 T y y Thereore y1 y 2 T y1, y 2, or all y1, y 2 2 (ii) We have xy x and xy y 1 We have x1 x 2 in y / y x1 x in x x / y x, y x in x / y x

13 Homomorphism on T nti-fuzzy Ideals O Near-ing 47 x2 Thereore x1 x 2 x 2, or all y1, y 2 2 and 1 (iii) we have y1 y 2 in x / x y1 y in x x / x y, x y x1 B x 2 x1 y1 x 2 y 2 in max, /, in T x, x / x y, x y T in x / x y, in x / x y 1, 2 T y y Thereore y1 y 2 T y1, y 2, or all y1, y (iv) we have y1 y 2 in x / x y1 y in x x / x y, x y x1 x 2 x1 y1 x 2 y 2 in max, /, in T x, x / x y, x y T in x / x y, in x / x y 1, 2 T y y Thereore y1 y 2 T y1, y 2, or all y1, y 2 2 Hence is a T anti-uzzy ideal o a ring 2.

14 48 J. Prakashmanimaran, B.Chellappa and M. Jeyakumar EFEENCES [1] M. kram, On T-uzzy ideals in near-rings, Int. J. Math. Math. Sci. Volume 2007 (2007), rticle ID 73514, 14 pages [2] M. T. bu Osman, On some product o uzzy subgroups, Fuzzy Sets and Systems 24 (1987), [3] W. E. Barnes, On the Γ-rings o Nobusawa, Paciic J. Math. 18 (1966), [4] G. L. Booth, note on near-rings, Studia. Sci. Math. Hungar. 23 (1988) [5] P. Deena and G. Mohanraj, T-uzzy ideals in rings, International Journal o Computational Cognition 2 (2011) [6] W. Liu, Fuzzy invariant subgroups and uzzy ideals, Fuzzy Sets and Systems 8 (1982) [7] T. Srinivas, T. Nagaiah and P. Narasimha Swamy, nti uzzy ideals o near-rings, nn. Fuzzy Math. Inorm. 3(2) (2012) [8] Y.-D. Yu and Z.-D. Wang. TL-subrings and TL-ideals part1: basic concepts. Fuzzy Sets and Systems, 68:93 103, [9] W. E. Coppage and J. Luh, adicals o gamma-rings, J. Math. Soc. Japan 23 (1971), [10] N. Nobusawa, On a generalization o the ring theory, Osaka J. Math. 1 (1964), [11] B. Schweizer and. Sklar, Statistical metric spaces, Paciic Journal o Mathematics. 10 (No. 1) (1963), [12] Y. Yu, J. N. Mordeson and S. C. Cheng, Elements o L-algebra, Lecture Notes in Fuzzy Math. and Computer Sciences, Creighton Univ., Omaha, Nebraska 68178, US (1994). [13] L.. Zadeh, Fuzzy sets, Inorm. and Control, 8 (1965),