On Fuzzy 2-absorbing Γ-ideals in Γ-rings

Yıl 2022, Cilt: 14 Sayı: 1, 32 - 43, 30.06.2022

Serkan Onar

https://doi.org/10.47000/tjmcs.907088

Öz

The goal of this paper is to give a definition of a generalization of fuzzy prime $\Gamma $-ideals in $\Gamma $-rings by introducing fuzzy 2-absorbing $\Gamma $-ideals and fuzzy weakly completely 2-absorbing $\Gamma $-ideals of commutative $\Gamma $-rings and to give their properties. Furthermore, we give a diagram which transition between definitions of fuzzy 2-absorbing $\Gamma $-ideals of a $\Gamma $-ring. Finally, we introduce fuzzy quotient $\Gamma $-ring of $R$ induced by the fuzzy weakly completely 2-absorbing $\Gamma $-ideal is a $2$-absorbing $\Gamma $-ring.

Anahtar Kelimeler

2-absorbing, fuzzy 2-absorbing ideal, fuzzy 2-absorbing Γ-ideal, fuzzy weakly completely 2-absorbing Γ-ideal, fuzzy K-2-absorbing Γ-ideal

Kaynakça

• Anderson, D.F., Badawi, A., On n-absorbing ideals of commutavie rings, Commutative Algebra, 39(2011), 1646-1672.
• Badawi, A., On 2-absorbing ideals of commutative rings, Bull. Austral Math. Soc., 75(2007), 417-429.
• Badawi, A., Tekir, U., Yetkin, E., On 2-absorbing primary ideals in commutative rings, Bull. Austral. Math. Soc., 51(4)(2014), 1163-1173.
• Badawi, A., Darani, A.Y., On weakly 2-absorbing ideals of commutative rings, Houston J. Math., 39(2013), 441-452.
• Barnes, W.E., On the $\Gamma -$rings of Nobusawa, Pacific J. Math., 18(1966), 411-422.
• Darani, A.Y., On $L-$fuzzy 2-absorbing ideals, Italian Journal of Pure and Appl. Math., 36(2016), 147-154.
• Darani, A.Y., Puczylowski, E.R., On 2-absorbing commutative semigroups and their applications to rings, Semigroup Forum, 86(2013), 83-91.
• Darani, A.Y., Hashempoor, A., $L-$fuzzy 0-(1- or 2- or 3-)2-absorbing ideals in semiring, Annals of Fuzzy Math. and Inform., 7(2)(2014), 303-311.
• Dutta, T.K., Chanda, T., Fuzzy prime ideals in $\Gamma -$rings, Bull. Malays. Math. Sci. Soc., 30(2007), 65-73.
• Dutta, T.K., Chanda, T., Structures of fuzzy ideals of $\Gamma-$ring, Bull. Malays. Math. Sci. Soc., 28(2005), 9-18.
• Elkettani, M.Y., Kasem, A., On 2-absorbing $\delta -$primary gamma ideal of gamma ring, Int. J. Pure and Appl. Math., 106(2)(2016), 543-550.
• Ersoy, B.A., Fuzzy semiprime ideals in $\Gamma -$rings, Int. J. Physical Sciences, 5(4)(2010), 308-312.
• Jun, Y.B., On fuzzy prime ideals of $\Gamma -$rings, Soochow J. Math., 21(1)(1995), 41-48.
• Kumar, P., Dubey, M.K., Sarohe, P., Some results on 2-absorbing ideals in commutative semirings, J. Math. and Appl., 38(2015), 77-84.
• Kuraoka, T., Kuroki, T., On fuzzy quotient rings induced by fuzzy ideals, Fuzzy Sets and Systems, 47(1992), 381-386.
• Kyuno, S., On prime gamma rings, Pacific J. Math. 75(1)(1978), 185-190.
• Kyuno, S., A gamma ring with the right and left unities, Math. Jopanica, 24(2)(1979), 191-193.
• Kyuno, S., Prime ideals in gamma rings, Pacific J. Math, 98(2)(1982), 375-379.
• Liu, W.J., Operation on fuzzy ideals, Fuzzy Sets and Systems, 11(1983), 31-41.
• Liu, X., Idealistic soft $\Gamma -$rings, Journal of Hyperstructures, 2(2)(2013), 136-150.
• Luh, J., On the theory of simple $\Gamma -$rings, Michigan Math. J., 16(1969), 65-75.
• Malik, D.S., Mordeson, J.N., Fuzzy Commutative Algebra, World Scientific Publishing, 1998.
• Mukherjee, T.K., Sen, M.K., Prime fuzzy ideals in rings, Fuzzy Sets and Systems, 32(1989), 345-350.
• Nobusawa, N., On a generalization of the ring theory, Osaka J. Math., 1(1964), 81-89.
• Rosenfeld, A., Fuzzy groups, J. Math. Anal. Appl., 35(1971), 512-517.
• Sönmez, D., Yeşilot, G., Onar, S., Ersoy, B.A., Davvaz, B., On 2-absorbing primary fuzzy ideals of commutative rings, Mathematical Problems in Engineering, (2017), Article ID 5485839.
• Zadeh, L.A., Fuzzy sets, Inform and Control, 8(1965), 338-353.