A study on ideal elements in ordered $\Gamma$-semirings

Rao, M. Murali Krishna  ; Rafi, Noorbhasha ; Kumar Kona, Rajendra

Discussiones Mathematicae. General Algebra and Applications, Tome 43 (2023) no. 2, pp. 249-261.

Résumé

The aim of this paper is to study the structures of some ordered semigroups not only with the ideal elements but also with the generalization of ideal elements. The ideal elements play an important and necessary role in studying the structure of ordered semigroups. We introduce the notion of (ideal, interior ideal, quasi ideal, bi-ideal, quasi interior ideal and weak interior ideal) elements of ordered Γ-semirings. We study the properties of ideal elements, relations between them and characterize the ordered Γ-semirings, regular ordered Γ-semirings and simple ordered Γ-semirings using ideal elements. We prove that if M be a simple ordered Γ-semiring, then every element of M is an ideal element of M.

Détail

Keywords: ideal elements, interior ideal elements, Bi-ideal elements, quasi interior ideal elements, weak interior ideal elements, ordered $\Gamma$-semirings

Bibliographie

[1] P.J. Allen, A fundamental theorem of homomorphism for semirings, Proc. Amer. Math. Sco. 21 (1969) 412–416.

[2] A.H. Cllliford and G.B. Preston, The Algebraic Theory of Semigroups Vol. I, Amer. Math. Soc. Math. Surveys 7 (Providence, RI, USA, 1961).

[3] R.A. Good and D.R. Hughes, Associated groups for a semigroup, Bull. Amer. Math. Soc. 58 (1952) 624–625.

[4] M. Henriksen, Ideals in semirings with commutative addition, Amer. Math. Soc. Noti 5 (1958) 321.

[5] K. Iseki, Quasi-ideals in semirings without zero, Proc. Japan Acad. 34 (1958) 79–84.

[6] K. Iseki, Ideal theory of semiring, Proc. Japan Acad. 32 (1956) 554–559.

[7] K. Iseki, Ideal in semirings, Proc. Japan Acad. 34 (1958) 29–31.

[8] K. Izuka, On the Jacobson radical of a semiring, Tohoku Math. J. 11 (2) (1959) 409–421.

[9] R.D. Jagatap and Y.S. Pawar, Quasi-ideals and minimal quasi-ideals in $Γ$-semirings, Novi Sad J. Math. 39 (2) (2009) 79–87.

[10] N. Kehayopulu, Interior ideals and interior ideal elements in ordered semigroups, Pure Math. Appl. 2 (3)(1999) 407–409.

[11] S. Lajos and F.A. Szasz, On the bi-ideals in associative ring, Proc. Japan Acad. 46 (1970) 505–507.

[12] S. Lajos, On the bi-ideals in semigroups, Proc. Japan Acad. 45 (1969) 710–712.

[13] S. Lajos, $(m;k;n)$-ideals in semigroups, in: Notes on Semigroups II, Karl Marx Univ. Econ. Dept. Math. Budapest (1976) (1) 12–19.

[14] H. Lehmer, A ternary analogue of abelian groups, Amer. J. Math. 59 (1932) 329–338.

[15] G. Lister, Ternary rings, Trans. Amer. Math. Soc. 154 (1971) 37–55. https://doi.org/10.2307/1995425

[16] M.M.K. Rao, $Γ$-semiring with identity, Discuss. Math. General Alg. and Appl. 37 (2017) 189–207. https://doi.org/10.7151/dmgaa.1276

[17] M.M.K. Rao, The Jacobson radical of $Γ$-semiring, South. Asian Bull. Math. 23 (1999) 127–134.

[18] M.M.K. Rao and B. Venkateswarlu, Regular $Γ$-incline and field $Γ$-semiring, Novi Sad J. Math. 19 (2015) 155–171. emis.ams.org/journals/NSJOM/Papers/45-2 /NSJOM-45-2-155-171.pdf.

[19] M.M.K. Rao, bi-interior ideals in semigroups, Discuss. Math. General Alg. and Appl. 38 (2018) 69–78. https://doi.org/10.7151/dmgaa.1283

[20] M.M.K. Rao, bi-interior ideals in $Γ$-semirings, Discuss. Math. General Alg. and Appl. 38 (2) (2018) 239–254. https://doi.org/10.7151/dmgaa.1283

[21] M.M.K. Rao, Ideals in ordered $Γ$-semirings, Discuss. Math. General Alg. and Appl. 38 (2018) 47–68. https://doi.org/10.7151/dmgaa.1284

[22] M.M.K. Rao, A study of generalization of bi- ideal, quasi- ideal and interior ideal of semigroup, Math. Morovica 22 (2) (2018) 103–115. https://doi.org/10.5937/MatMor1802103M

[23] M.M.K. Rao, A study of bi-quasi-interior ideal as a new generalization of ideal of generalization of semiring, Bull. Int. Math. Virtual Inst. 8 (2018) 519–535. https://doi.org/10.7251/JIMVI1801019R

[24] M.M.K. Rao, A study of quasi-interior ideal of semiring, Bull. Int. Math. Virtual Inst. 2 (2019) 287–300. https://doi.org/10.7251/BIMVI1902287M

[25] M.M.K. Rao, Left bi-quasi ideals of semirings, Bull. Int. Math. Virtual Inst. 8 (2018) 45–53. https://doi.org/10.7251/BIMVI1801045R

[26] M.M.K. Rao, B. Venkateswarlu and N. Rafi, Left bi-quasi-ideals of $Γ$-semirings, Asia Pacific J. Math. 4 (2) (2017) 144–153.

[27] M.M.K. Rao, B. Venkateswarlu and N. Rafi, On r-ideals of $Γ$-inclines, Asia Pacific J. Math. 5 (2) (2019) 208–218. https://doi.org/10.28924/APJM/6-6

[28] M.M.K. Rao, Bi-quasi-ideals and fuzzy bi-quasi ideals of $Γ$- semigroups, Bull. Int. Math. Virtual Inst. 7 (2) (2017) 231–242. imvibl.org /JOURNALS / BULLETIN.

[29] M.M.K. Rao, $Γ$-semirings-I, South. Asian Bull. Math. 19 (1) (1995) 49–54.

[30] M.M.K. Rao, B. Venkateswarlu and N. Rafi, Fuzzy $r$-ideals in $Γ$-incline, Ann. Fuzzy Math. Inform. 13 (2) (2019) 253–276. https://doi.org/10.30948/afmi.2019.17.3.247

[31] N. Nobusawa, On a generalization of the ring theory, Osaka. J. Math. 1 (1964) 81–89. ir.library.osaka-u.ac.jp/dspace/bitstream/11094/12354/1/ojm01-01-08.pdf.

[32] M.K. Sen, On $Γ$-semigroup, in: Proc. of International Conference of algebra and its application (Decker Publicaiton, New York, 1981) 301–308.

[33] A.M. Shabir and A. Batod, A note on quasi ideal in semirings, South. Asian Bull. Math. 7 (2004) 923–928.

[34] O. Steinfeld, Uher die quasi ideals, Von halbgruppend, Publ. Math. Debrecen 4 (1956) 262–275.

[35] G. Szasz, Interior ideals in semigroups, in: Notes on semigroups IV, KarlMarx Univ. Econ. Dept. Math. Budapest 5 (1977) 1–7.

[36] G. Szasz, Remark on interior ideals of semigroups, Studia Scient. Math. Hung. 16 (1981) 61–63.

[37] 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. https://doi.org/10.1090/S0002-9904-1934-06003-8