Prime bi-interior ideals of Γ-semirings

Document Type : Research Paper

Author

Department of Mathematics, Sankethika Engineering College , Visakhapatnam, India

Abstract

In this paper, we introduce the notion of prime bi-interior ideal, bi-interior ideal,strongly prime bi-interior ideal,semiprime strongly irreducible bi-interior ideal and irreducible bi-interior ideal.we study these ideals properties and relation between them and also characterize regular Γ-semiring and Γ-semiring using prime bi-interior ideals.
 

Keywords


[1] T. K. Dutta and S. K. Sardar, Semi-prime Ideals and Irreducible Ideals of Γ- semirings, Novi Sad.J.Math.,1 (2000), 97-108.
[2] R. A. Good and D. R. Hughes, Associated groups for a semigroup, Bull. Amer. Math. Soc., 58 (1952), 624-625.
[3] M. Henriksen, Ideals in semirings with commutative addition, Amer. Math. Soc. Notices, 5 (1958), 321.
[4] K. Iseki, Ideal Theory of Semiring, Proc.Japan Acad., 32 (8)(1956), 554-559.
[5] S. Lajos, On the bi-ideals in semigroups, Proc. Japan Acad., 45 (1969), 710-712.
30 M. Murali Krishna Rao
[6] S. Lajos and F. A. Szasz, On the bi-ideals in associative ring, Proc. Japan Acad.,
[7] M. Shabir, A. Ali and S. A. Batool, Note On Quasi-ideals in Semirings, Southeast Asian Bulletin of Mathematics, 27 (2004), 923-928.
[8] M. Shabir and N. Kanwal, Prime Bi-ideals in Semigroups, Southeast Asian Bulletin of Mathematics, 31 (2007), 757-764.
[9] M. Murali Krishna Rao, Γ-semiring-I, Southeast Asian Bull. Math. 19 (1)(1995), 49-54.
[10] M. Murali Krishna Rao, Γ-semiring-II, Southeast Asian Bulletin of Mathematics, 21(3)(1997), 281-287
[11] M.Murali Krishna Rao, The Jacobson radical of Γ-semiring, South eastAsian Bulletin of Mathematics, 23 (1999), 127-134
[12] M. Murali Krishna Rao, Γ-semiring with identity, Discussiones Mathematicae General Algebra and Applications, 37 (2017), 189-207.
[13] M. Murali Krishna Rao, Γ-semiring with identity, Discussiones Mathematicae General Algebra and Applications. , 37 (2017) 189-207.
[14] M.Murali Krishna Rao, Ideals in ordered Γ-semirings, Discussiones Mathematicae General Algebra and Applications 38 (2018), 47-68.
[15] M. Murali Krishna Rao, Left bi-quasi ideals of semirings, Bull. Int. Math. Virtual Inst, 8 (2018), 45-53.
[16] M. Murali Krishna Rao, B. Venkateswarlu and N. Ra , Left bi-quasi-ideals of Γ-semirings, Asia Paci c Journal of Mathematics, 4(2) (2017), 144-153.
[17] M. Murali Krishna Rao and B. Venkateswarlu Bi-interior ideals of Γ-semirings, Discussiones Mathematicae General Algebra and Applications, 38(2) (2018), 239-254.
[18] N. Nobusawa, On a generalization of the ring theory, Osaka. J.Math., 1 (1964), 81-89.
[19] O. Steinfeld, Uher die quasi ideals, Von halbgruppend Publ. Math., Debrecen, 4 (1956), 262-275.
[20] 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.