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Journal of Discrete Mathematical Sciences and Cryptography

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Journal of Discrete Mathematical Sciences and Cryptography
Open Access Research Article

On commutativity of prime Γ-rings

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pp. 1971–1976Vol. 29Issue 5May 2026DOI: 10.47974/JDMSC-2377 Crossmark XML
Received:
01 May 2025
Published Online:
22 May 2026
Article type:
Research Article
Language:
EN
Article no.:
JDMSC-2377
Pages:
1971–1976

Abstract

In the context of automorphism σ defined on a prime Γ-ring Q, we explored various results related to commutativity within Q by examining a generalized reverse σ-derivations Ħ and Ɠ of Q that satisfy any one of the properties: (i)    If Ħ(Q) ⊂ 𝖹(Q). (ii)   If [Ħ(r), Ɠ(s)]α = 0 and Ħ is commuting. (iii)  If Ɠ is commuting s.t. (Ħ(r)αs + Ħ(s)αr) + (rαƓ(s) + sαƓ(r)) = 0. (iv)  If [Ħ(r), s]α = [Ħ(s), r]α and Ħ is commuting. For each r, s belongs to Q and 𝛼 in Γ, hence Q is commutative. This study aims to analyze the commutativity of the primitive gamma ring (Γ-ring) by introducing the concept of generalized reverse σ-derivatives associated with the automorphism σ. Also explore the properties of two of these derivatives, Ħ and Ɠ, and investigate the conditions that guarantee Γ-ring commutativity. The main results include an analysis of the relationship between these derivatives and their influence on the Γ-ring structure, while generalizing previous results in this field. The study provides a comprehensive theoretical framework for a deeper understanding of the behavior of these Γ-rings under the influence of generalized reverse σ-derivatives, opening new research horizons in the theory of non-commutative rings.

Keywords

Prime Γ-ringσ-derivationReverse σ-derivationGeneralized reverse σ-derivation

Subject Classifications

13D09

References

[1] N. Nobusawa, “On a generalization of the ring theory,” Osaka Journal of Mathematics, vol. 1, no. 1, pp. 81–89 (1964). 
[2] E. A. Ugurlu, “Generalizations of r-ideals of commutative rings,” Journal of Interdisciplinary Mathematics, vol. 24, no. 8, pp. 2283–2293 (2021). 
[3] S. Huang, “Generalized reverse derivations and commutativity of prime rings,” Communications in Mathematics, vol. 27, no. 1, pp. 43–50 (2019). 
[4] S. Kyuno, “On prime gamma rings,” Pac. J. Math., vol. 75, no. 1, pp. 185–190 (1978). 
[5] S. A. Hamil, “Generalized reverse derivations and commutativity of prime Γ-semirings,” J. Phys.: Conf. Ser, vol. 1804, no. 1, p. 12092 (2021). 
[6] S. Huang and S. Ali, “The commutativity of prime Γ-rings with generalized skew derivations,” Georgian Mathematical Journal, vol. 24, no. 3, pp. 393–402 (2017). 
[7] T. K. Lee and J. H. Lin, “Jordan τ-derivations of prime rings,” Commun. Algebra, vol. 43, no. 12, pp. 5195–5204 (2015). 
[8] K. K. Dey, A. C. Paul, and I. S. Rakhimov, “Semiprime gamma rings with orthogonal reverse derivations,” International Journal of Pure and Applied Mathematics, vol. 83, no. 2, pp. 233–245 (2013). 
[9] M. Ibraheem, “Reverse derivations on prime gamma near rings,” Int. J. Pure Appl. Sci. Technol., vol. 26, no. 2, p. 64 (2015). 
[10] M. A. Öztürk, Y. B. Jun, and K. H. Kim, “On derivations of prime gamma rings,” Turkish Journal of Mathematics, vol. 26, no. 3, pp. 317–328 (2002). 

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