On commutativity of quotient semirings through generalized derivations

Tariq Mahmood; Liaqat Ali; Muhammad Aslam; Ghulam Farid; Tariq Mahmood; Liaqat Ali; Muhammad Aslam; Ghulam Farid

doi:10.3934/math.20231312

AIMS Mathematics

2023, Volume 8, Issue 11: 25729-25739. doi: 10.3934/math.20231312

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Research article

On commutativity of quotient semirings through generalized derivations

1.
Department of Mathematics, G.C University Lahore, Pakistan
2.
Department of Mathematics, Govt. MAO Graduate College Lahore, Pakistan
3.
Department of Mathematics, G.C University Lahore, Pakistan
4.
Department of Mathematics, COMSATS University Islamabad, Attock Campus, Pakistan

Received: 10 March 2023 Revised: 16 June 2023 Accepted: 24 July 2023 Published: 06 September 2023
MSC : 16N60, 16U80, 16W25, 16Y60

This research article aims to study quotient MA-semirings determined by the prime ideals. Derivations are important tools to study algebraic structures. We establish some theorems on commutativity of quotient MA-semirings under certain differential identities. Results of this paper are extensions of many well known facts of this topic.
- MA-semirings,
- generalized derivations,
- prime ideals,
- $ Q $-ideals
Citation: Tariq Mahmood, Liaqat Ali, Muhammad Aslam, Ghulam Farid. On commutativity of quotient semirings through generalized derivations[J]. AIMS Mathematics, 2023, 8(11): 25729-25739. doi: 10.3934/math.20231312

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Abstract

This research article aims to study quotient MA-semirings determined by the prime ideals. Derivations are important tools to study algebraic structures. We establish some theorems on commutativity of quotient MA-semirings under certain differential identities. Results of this paper are extensions of many well known facts of this topic.

References

[1]	M. A. Javed, M. Aslam, M. Hussain, On condition (A$_{2}$) of Bandlet and Petrich for inverse semirings, Int. Math. Forum, 7 (2012), 2903–2914.
[2]	H. J. Bandlet, M. Petrich, Subdirect products of rings and distrbutive lattics, Proc. Edinb. Math. Soc., 25 (1982), 135–171. https://doi.org/10.1017/S0013091500016643 doi: 10.1017/S0013091500016643
[3]	S. Sara, M. Aslam, M. A. Javed, On centralizer of semiprime inverse semiring, Discuss. Math. Gen. Algebra Appl., 36 (2016), 71–84.
[4]	Y. A. Khan, M. Aslam, L. Ali, Commutativity of inverse semirings through f(xy) = [x, f(y)], Thai J. Math., 2018 (2018), 288–300.
[5]	L. Ali, M. Aslam, M. I. Qureshi, Y. A. Khan, S. Rehman, G. Farid, Commutativity of MA-semirings with involution through generalized derivations, J. Math., 2020 (2020), 8867247. https://doi.org/10.1155/2020/8867247 doi: 10.1155/2020/8867247
[6]	L. Ali, M. Aslam, G. Farid, S. A. Khalek, On differential identities of Jordan ideals of semirings, AIMS Mathematics, 6 (2020), 6833–6844. http://doi.org/10.3934/math.2021400 doi: 10.3934/math.2021400
[7]	L. Ali, Y. A. Khan, A. A. Mousa, S. A. Khalek, G. Farid, Some differential identities of MA-semirings with involution, AIMS Mathematics, 6 (2020), 2304–2314. http://doi.org/2010.3934/math.2021139
[8]	S. E. Atani, The zero-divisor graph with respect to ideals of a commutative semiring, Glas. Math., 43 (2008), 309–320. https://doi.org/10.3336/gm.43.2.06 doi: 10.3336/gm.43.2.06
[9]	S. E. Atani, R. E. Atani, Some remarks on partitioning semirings, An. St. Univ. Ovidius Constanta, 18 (2010), 49–62.
[10]	K. Iséki, Quasiideals in semirings without zero, Proc. Jpn. Acad., 34 (1958), 79–84. https://doi.org/10.3792/pja/1195524783 doi: 10.3792/pja/1195524783
[11]	M. K. Sen, M. R. Adhikari, On k-ideals of semirings, Int. J. Math. Math. Sci., 15 (1992), 642431. https://doi.org/10.1155/S0161171292000437 doi: 10.1155/S0161171292000437
[12]	R. Awtar, Lie and Jordan structure in prime rings with derivations, Proc. Amer. Math. Soc., 41 (1973), 67–74.
[13]	H. E. Mir, A. Mamouni, L. Oukhtite, Commutativity with algebraic identities involving prime ideals, Commun. Korean Math. Soc., 35 (2020), 723–731. https://doi.org/10.4134/CKMS.c190338 doi: 10.4134/CKMS.c190338
[14]	M. Bresar, On the distance of the composition of two derivations to the generalized derivations, Glasg. Math. J., 33 (1991), 89–93. https://doi.org/10.1017/S0017089500008077 doi: 10.1017/S0017089500008077
[15]	J. Berger, I. N. Herstein, J. W. Kerr, Lie ideals and derivations of prime rings, J. Algebra, 71 (1981), 259–267.
[16]	H. E. Bell, W. S. Martindale, Centralizing mappings of semiprime rings, Canad. Math. Bull., 30 (1987), 92–101. https://doi.org/10.4153/CMB-1987-014-x doi: 10.4153/CMB-1987-014-x
[17]	D. A. Jordan, On the ideals of a Lie algebra of derivations, J. Lond. Math. Soc., s2–33 (1986), 33–39. https://doi.org/10.1112/jlms/s2-33.1.33 doi: 10.1112/jlms/s2-33.1.33
[18]	E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc., 8 (1957), 1093–1100.
[19]	N. Ur Rehman, H. M. Alnoghashi, Action of prime ideals on generalized derivations-I, arXiv, 2021. https://doi.org/10.48550/arXiv.2107.06769

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