Generalized (τ,σ)-L-Derivations in Rings

Saber, Hicham; Al-Amery, Zakia Z.; Al-omary, Radwan M.; Aldwoah, Khaled; Alsulami, Amer; Suhail, Muntasir

doi:10.3390/math13172784

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Open AccessArticle

Generalized (τ,σ)-L-Derivations in Rings

by

Hicham Saber

¹,

Zakia Z. Al-Amery

²,

Radwan M. Al-omary

³

,

Khaled Aldwoah

^4,*

,

Amer Alsulami

⁵ and

Muntasir Suhail

^6,*

¹

Department of Mathematics, College of Science, University of Ha’il, Ha’il 55473, Saudi Arabia

²

Department of Mathematics, Aden University, Aden 5243, Yemen

³

Department of Mathematics, Ibb University, Ibb 70270, Yemen

⁴

Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah 42351, Saudi Arabia

⁵

Department of Mathematics, Turabah University College, Taif University, Taif 21944, Saudi Arabia

⁶

Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia

^*

Authors to whom correspondence should be addressed.

Mathematics 2025, 13(17), 2784; https://doi.org/10.3390/math13172784

Submission received: 23 July 2025 / Revised: 19 August 2025 / Accepted: 26 August 2025 / Published: 29 August 2025

(This article belongs to the Section A: Algebra and Logic)

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Abstract

Let

τ

and

σ : X ⟶ X

be automorphisms of an arbitrary associative ring X, and let L be a prime ideal of X. The main objective of this article is to combine the notions of generalized L-derivations and

(τ, σ)

-L-derivations by introducing and analyzing a novel additive mapping

Π : X \to X

called a generalized

(τ, σ)

-L-derivation associated with a

(τ, σ)

-L-derivation

π

. Later, we will examine the algebraic properties of a factor ring

X / L

under the influence of certain algebraic expressions containing this generalized

(τ, σ)

-L-derivation and lying in a prime ideal L. Through our main findings, we establish certain results under different conditions. It also provides various illustrative examples to show that our primeness hypotheses in various theorems are not exaggerated.

Keywords: (τ,σ)-P-derivation; generalized (τ,σ)-P-derivation; integral domain; prime ideal; commutativity; mathematical operators; factor ring

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MDPI and ACS Style

Saber, H.; Al-Amery, Z.Z.; Al-omary, R.M.; Aldwoah, K.; Alsulami, A.; Suhail, M. Generalized (τ,σ)-L-Derivations in Rings. Mathematics 2025, 13, 2784. https://doi.org/10.3390/math13172784

AMA Style

Saber H, Al-Amery ZZ, Al-omary RM, Aldwoah K, Alsulami A, Suhail M. Generalized (τ,σ)-L-Derivations in Rings. Mathematics. 2025; 13(17):2784. https://doi.org/10.3390/math13172784

Chicago/Turabian Style

Saber, Hicham, Zakia Z. Al-Amery, Radwan M. Al-omary, Khaled Aldwoah, Amer Alsulami, and Muntasir Suhail. 2025. "Generalized (τ,σ)-L-Derivations in Rings" Mathematics 13, no. 17: 2784. https://doi.org/10.3390/math13172784

APA Style

Saber, H., Al-Amery, Z. Z., Al-omary, R. M., Aldwoah, K., Alsulami, A., & Suhail, M. (2025). Generalized (τ,σ)-L-Derivations in Rings. Mathematics, 13(17), 2784. https://doi.org/10.3390/math13172784

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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