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Journals / Fundamentals of Contemporary Mathematical Sciences / 2025 / Volume: 6 Issue: 2

Semiprime Ideal of Rings with Symmetric Bi-Derivations

Emine Koç Sögütcü(SİVAS CUMHURİYET ÜNİVERSİTESİ)0000-0002-8328-4293
Oznur Golbasi(SİVAS CUMHURİYET ÜNİVERSİTESİ)0000-0002-9338-6170
Pages
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Publication Date
2025-07-30
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AI Summary & Key Takeaways

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⚡ TL;DR(Too Long; Didn't Read)

In the present paper, we shall prove that δ is a ℘-commuting map on I if any one of the following holds: i. δ(σ) ○ κ ∈ ℘, ii. δ([σ, κ]) ± [δ(σ), κ] ∈ ℘, iii. δ(σ○κ)±(δ(σ)○κ) ∈ ℘, iv. δ([σ, κ])±δ(σ)○κ ∈ ℘, v. δ(σ○κ)±[δ(σ), κ] ∈ ℘, vi. δ(σ)○κ±[δ(κ), σ] ∈ ℘, vii. δ([σ, κ]) ± δ(σ) ○ κ − [δ(κ), σ] ∈ ℘, viii. δ([σ, κ]) ± [δ(σ), κ] + [δ(κ), σ] ∈ ℘, ix. Δ(σ, κκ3) ±Δ(σ, κ)κ3 ∈ ℘, x. Δ(δ(σ), σ) ∈ ℘, xi. δ(δ(σ)) = g(σ), xii. δ(σ)κ ± σg(κ) ∈ ℘, xiii. [δ(σ), κ] ± [g(κ), σ] ∈ ℘, xiv. δ(σ) ○ κ ± (σ ○ g(κ)) ∈ ℘, xv. [δ(σ), κ] ± (σ ○ g(κ)) ∈ ℘, xvi. δ(σ) ○ κ ± [g(κ), σ] ∈ ℘ for all σ, κ ∈ I where G ∶ ℵ × ℵ → ℵ is a symmetric bi-derivation such that g is the trace of G.

📌 3 Key Takeaways

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Objective & Hypothesis:Let Ω be a ring with ℘ a semiprime ideal of Ω, I an ideal of Ω, Δ ∶ Ω × Ω → Ω a symmetric bi-derivation and δ be the trace of Δ.
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Key Findings & Data:Let Ω be a ring with ℘ a semiprime ideal of Ω, I an ideal of Ω, Δ ∶ Ω × Ω → Ω a symmetric bi-derivation and δ be the trace of Δ.
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Conclusion & Impact:In the present paper, we shall prove that δ is a ℘-commuting map on I if any one of the following holds: i. δ(σ) ○ κ ∈ ℘, ii. δ([σ, κ]) ± [δ(σ), κ] ∈ ℘, iii. δ(σ○κ)±(δ(σ)○κ) ∈ ℘, iv. δ([σ, κ])±δ(σ)○κ ∈ ℘, v. δ(σ○κ)±[δ(σ), κ] ∈ ℘, vi. δ(σ)○κ±[δ(κ), σ] ∈ ℘, vii. δ([σ, κ]) ± δ(σ) ○ κ − [δ(κ), σ] ∈ ℘, viii. δ([σ, κ]) ± [δ(σ), κ] + [δ(κ), σ] ∈ ℘, ix. Δ(σ, κκ3) ±Δ(σ, κ)κ3 ∈ ℘, x. Δ(δ(σ), σ) ∈ ℘, xi. δ(δ(σ)) = g(σ), xii. δ(σ)κ ± σg(κ) ∈ ℘, xiii. [δ(σ), κ] ± [g(κ), σ] ∈ ℘, xiv. δ(σ) ○ κ ± (σ ○ g(κ)) ∈ ℘, xv. [δ(σ), κ] ± (σ ○ g(κ)) ∈ ℘, xvi. δ(σ) ○ κ ± [g(κ), σ] ∈ ℘ for all σ, κ ∈ I where G ∶ ℵ × ℵ → ℵ is a symmetric bi-derivation such that g is the trace of G.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Rings · ideals · semiprime ideals

Abstract

Let Ω be a ring with ℘ a semiprime ideal of Ω, I an ideal of Ω, Δ ∶ Ω × Ω → Ω a symmetric bi-derivation and δ be the trace of Δ. In the present paper, we shall prove that δ is a ℘-commuting map on I if any one of the following holds: i. δ(σ) ○ κ ∈ ℘, ii. δ([σ, κ]) ± [δ(σ), κ] ∈ ℘, iii. δ(σ○κ)±(δ(σ)○κ) ∈ ℘, iv. δ([σ, κ])±δ(σ)○κ ∈ ℘, v. δ(σ○κ)±[δ(σ), κ] ∈ ℘, vi. δ(σ)○κ±[δ(κ), σ] ∈ ℘, vii. δ([σ, κ]) ± δ(σ) ○ κ − [δ(κ), σ] ∈ ℘, viii. δ([σ, κ]) ± [δ(σ), κ] + [δ(κ), σ] ∈ ℘, ix. Δ(σ, κκ3) ±Δ(σ, κ)κ3 ∈ ℘, x. Δ(δ(σ), σ) ∈ ℘, xi. δ(δ(σ)) = g(σ), xii. δ(σ)κ ± σg(κ) ∈ ℘, xiii. [δ(σ), κ] ± [g(κ), σ] ∈ ℘, xiv. δ(σ) ○ κ ± (σ ○ g(κ)) ∈ ℘, xv. [δ(σ), κ] ± (σ ○ g(κ)) ∈ ℘, xvi. δ(σ) ○ κ ± [g(κ), σ] ∈ ℘ for all σ, κ ∈ I where G ∶ ℵ × ℵ → ℵ is a symmetric bi-derivation such that g is the trace of G.

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Cite This Article

Sögütcü, E. K., Golbasi, O. (2025). Semiprime Ideal of Rings with Symmetric Bi-Derivations. Fundamentals of Contemporary Mathematical Sciences, Volume: 6 Issue: 2. https://doi.org/10.54974/fcmathsci.1643355

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References (12)

  1. Ashraf M., On symmetric bi-derivations in rings, Rendiconti dell’Istituto di Matematica dell’Universit`a di Trieste, 31, 25-36, 1999.
  2. Ashraf M., Rehman N., On derivations and commutativity in prime rings, East-West Journal of Mathematics, 3(1), 87-91, 2001.
  3. Ashraf M., Ali A., Rani R., On generalized derivations of prime rings, Southeast Asian Bulletin of Mathematics, 29, 669-675, 2005.
  4. Çelik M., Koç Söğütçü E., Multiplicative generalized derivations acting on the semiprime ideals of the ring, International Journal of Open Problems in Computer Science and Mathematics, 17(3), 44-56, 2024.
  5. Daif M.N., Bell H.E., Remarks on derivations on semiprime rings, International Journal of Mathematics and Mathematical Sciences, 15(1), 205-206, 1992.
  6. Koç Söğütçü E., Gölbaşı Ö., Some results on Lie ideals with symmetric reverse bi-derivations in semiprime rings I, Facta Universitatis, Series: Mathematics and Informatics, 36(2), 309-319, 2021.
  7. Koç Söğütçü E., Gölbaşı Ö., Ünalan H., Some results on Lie ideals with symmetric reverse biderivations in semiprime rings II, Bulletin of the International Mathematical Virtual Institute, 12(3), 477-485, 2022.
  8. Maksa G., A remark on symmetric biadditive functions having nonnegative diagonalization, Glasnik Matematicki, 15(35), 279-282, 1980.
  9. Maksa G., On the trace of symmetric bi-derivations, Comptes Rendus Math´ematique Representation Acad´emie des Sciences Canada, 9, 303-307, 1987.
  10. Posner E.C., Derivations in prime rings, Proceedings of the American Mathematical Society, 8, 1093- 1100, 1957.
  11. Reddy C.J., Rao G.V., Reddy K.M., Symmetric left bi-derivations in semiprime rings, IOSR Journal of Mathematics, 11(5), 25-26, 2015.
  12. Vukman J., Two results concerning symmetric bi-derivations on prime rings, Aequationes Mathematicae, 40, 181-189, 1990.