EXTENSIONS OF FUZZY IDEALS OF Gamma- SEMIRINGS

Year 2018, Volume: 1 Issue: 3 - To memory of Prof. RNDr. Beloslav Rieˇcan, DrSc., 269 - 282, 24.10.2018

B Venkateswarlu , M Murali Krishna Rao , Y Adinarayana

Abstract

In this paper, we introduce the notion of extensions of fuzzy ideals of $\Gamma -$semiring, fuzzy weakly completely prime ideals and fuzzy $3-$weakly completely prime ideal of $\Gamma -$semiring. We study the relationship between fuzzy weakly completely prime ideals, fuzzy $3-$weakly prime ideals in terms of the extension of fuzzy ideals of $\Gamma -$semring.

Keywords

extension of a fuzzy ideal, fuzzy weakly completely prime ideal, fuzzy $3-$weakly completely prime ideal, Gamma semiring.

References

• \bibitem{1} P. J. Allen, A fundamental theorem of homomorphism for semirings, Proc. Amer. Math. Soc., 21 (1969), 412-416.\bibitem{9a} M. L. Das and T. K. Dutta, Extentsions of fuzzy ideals of semiring, Annalas of Fuzzy Math. and Info., 12 (5) (2016), 679-�690.\bibitem{3} T. K. Dutta and S. Kar, On regular ternary semirings, Advances in algebra, Proceedings of the ICM satellite conference in algebra and related topics, World scientific, (2003) 343-355.\bibitem{4} Y. B. Jun and C. Y. Lee, Fuzzy $\Gamma -$rings, Pusan kyongnan Math. Journal., 8 (1991), 163-170. \bibitem{5} N. Kuroki, On fuzzy semigroups, Information Sci., 53(3) ~(1991), 203--236. \bibitem{6} K. H. Kim, On right derivation of incline algebras, J. of chung cheng Math. Soc., 26 (4) (2013), 683--690.\bibitem{8} H. Lehmer, A ternary analogue of abelian groups, Amer. J. of Math., 59 (1932), 329--338.\bibitem{9} W. J. Liu, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy sets and Systems, 8 (2) (1982), 133--139.
• \bibitem{10} W. G. Lister, Ternary rings, Tran. of Amer. Math. Soc., 154 (1971), 37--55.\bibitem{11} D. Mandal, Fuzzy ideals and fuzzy interior ideals in ordered semirings, Fuzzy Info. and Engg., 6 (2014), 101--114.\bibitem{12} M. Murali Krishna Rao, Fuzzy soft $\Gamma-$semiring and fuzzy soft $k-$ideal over $\Gamma -$semiring, Annl. of Fuzzy Math. and Info., 9 (2)(2015), 12--25.
• \bibitem{13} M. Murali Krishna Rao, $\Gamma -$semirings-I, Southeast Asian Bull. of Math., 19 (1) (1995), 49--54.
• \bibitem{14} M. Murali Krishna Rao and B. Venkateswarlu, Regular $\Gamma -$semiring and field $\Gamma -$semiring, Novi Sad J. of Math., 45 (2) (2015), 155--171.\bibitem{14a} M. Murali Krishna Rao and B. Venkateswarlu, On fuzzy k-ideals, k-fuzzy ideals and fuzzy 2-prime ideals in $\Gamma -$semirings, J. Appl. Math. and Info., 34 (5) (2016), 405 - 419.\bibitem{15} V. Neumann, On regular rings, Proc. Nat. Acad. Sci., 22 (1936), 707--713.\bibitem{16} N. Nobusawa, On a generalization of the ring theory, Osaka. J.Math., 1 (1964), 81 -- 89.\bibitem{17} E . C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc., 8 (1957), 1093-1100. \bibitem{18} A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl., 35(1971), 512 -- 517.\bibitem{18a} S. K . Sardar and S. Goswami, Characterization of fuzzy prime ideals of $\Gamma -$semirings via operator semirings, Int. J. of Alg., 4(18) (2010), 867-873.\bibitem{19} M. K. Sen, On $\Gamma -$semigroup, Proc. of Inter. Conf. of Alg. and its Appl., Decker Publicaiton, New York, (1981), 301--308. \bibitem{20} U. M. Swamy and K. L. N. Swamy, Fuzzy prime ideals of rings, J. Math. Anal. Appl., 134 (1988), 94-- 103. \bibitem{21} H. S. Vandiver, Note on a simple type of algebra in which cancellation law of addition does not hold,Bull. Amer. Math. Soc., (N.S.), 40 (1934), 914--920. \bibitem{22} L. A. Zadeh, Fuzzy sets, Information and control, 8 (1965), 338--353.