GENERALIZATION OF ($\epsilon $, $\epsilon $ $\vee$ q)−FUZZY SUBNEAR-RINGS AND IDEALS

Document Type : Research Paper

Authors

Department of Mathematics, Annamalai University, Annamalainagar- 608002, India

Abstract

In this paper, we introduce the notion of ($\epsilon $, $\epsilon $ $\vee$ q_{k})fuzzy subnear-ring which is a generalization of ($\epsilon $, $\epsilon $ $\vee$ q)fuzzy subnear-ring. We have given examples which are ($\epsilon $, $\epsilon $ $\vee$ q_{k})fuzzy ideals but they are not ($\epsilon $, $\epsilon $ $\vee$ q)fuzzy ideals. We have also introduced the notions of ($\epsilon $, $\epsilon $ $\vee$ q_{k})fuzzyquasi-ideals and ($\epsilon $, $\epsilon $ $\vee$ q_{k})fuzzy bi-ideals of near-ring. We have characterized($\epsilon $, $\epsilon $ $\vee$ q_{k})fuzzy quasi-ideals and ($\epsilon $, $\epsilon $ $\vee$ q_{k})fuzzy bi-ideals of nearrings.

Keywords


[1] S. Abou-Zaid,On fuzzy subnear-rings and ideals, Fuzzy Sets and Systems, 44 (1991), 139-146.
[2] S. K. Bhakat and P. Das, On the definition of a fuzzy subgroup, Fuzzy Sets and Systems, 51
(1992), 235-241.
[3] S. K. Bhakat and P. Das, ($epsilon $, $epsilon $ $vee$ q)−fuzzy subgroup, Fuzzy Sets and Systems, 80 (1996),359-368.
[4] S. K. Bhakat and P. Das,Fuzzy subrings and ideals redefined, Fuzzy Sets and Systems, 81(1996), 383-393.
[5] S. K. Bhakat, ($epsilon $, $epsilon $ $vee$ q)−fuzzy normal, quasinormal and maximal subgroups, Fuzzy Sets and Systems, 112 (2000), 299-312.
[6] P. S. Das,Fuzzy groups and level subgroups, J. Math. Anal. Appl., 84 (1981), 264-269.
[7] V. N. Dixit, R. Kumar and A. Ajmal,Fuzzy ideals and fuzzy prime ideals of a ring, Fuzzy
Sets and Systems,4 (1991), 127-138.
[8] T. K. Dutta and B. K. Biswas,Fuzzy ideal of a near-ring, Bull. Cal. Math. Soc., 89 (1997),447-456.
[9] S. D. Kim and H. S. Kim,On fuzzy ideals of near-rings, Bull. Korean. Math. Soc., 33 (1996),593-601.
[10] R. Kumar,Fuzzy algebra, University of Delhi Publication Division, Delhi, Vol. I, 1993.
[11] N. Kuroki,Regular fuzzy duo rings, Inform. Sci., 94 (1996), 119-139.
[12] W. J. Liu,Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets and Systems, 8 (1982),
133-139.
[13] P. P Ming and L. Y. Ming,Fuzzy topology. I. neighborhood structure of a fuzzy point and
moore-smith convergence, J. Math. Anal. Appl., 76 (1980), 571-599.
[14] T. K. Mukherjee and M. K. Sen,Prime fuzzy ideals in rings, Fuzzy Sets and Systems, 32
(1989), 337-341.
[15] Al. Narayanan and T. Manikantan, ($epsilon $, $epsilon $ $vee$ q)−fuzzy subnear-rings and (2, 2 _q)fuzzy ideals of near-rings, J. Appl. Math. & Computing, 18 (2005), 419-430.
[16] G. Pilz,Near-rings, North-HollandMathematics Studies, 2nd ed., Vol. 23, North-Holland,
Amsterdam,1983.
[17] A. Rosenfeld,Fuzzy groups, J. Math. Anal. Appl., 35 (1971), 512-517.
[18] H. K. Saikia and L. K. Barthakur,On fuzzy Nsubgroups and fuzzy ideals of near-rings and
near-ring groups, The Journal of Fuzzy Mathematics, 11 (2003), 567-580.
[19] H. K. Saikia and L. K. Barthakur,Characterization of fuzzy substructures of a near-ring and
a near-ring group, The Journal of Fuzzy Mathematics, 13 (2005), 159-167.
[20] U. M. Swamy and K. L. N. Swamy,Fuzzy prime ideals of rings, J. Math. Anal. Appl., 134(1988), 94-103.
[21] T. Tamizh Chelvam and N. Ganesan,On bi-ideals of near-ring, Indian J. Pure Appl. Math.,
18(1987), 1002-1005.
[22] X. Y. Xie,On prime, quasi-prime, weakly quasi-prime fuzzy left ideals of semigroups, Fuzzy
Sets and Systems,123 (2001), 239-249.
[23] Z. Yue,Prime L-fuzzy ideals and primary L-fuzzy ideals, Fuzzy Sets and Systems, 27 (1988),
345-350.
[24] L. A. Zadeh,Fuzzy sets, Information and Control, 8 (1965), 338-353.