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All Journal Jambura Journal of Mathematics Euler : Jurnal Ilmiah Matematika, Sains dan Teknologi
Fitriani Fitriani
Universitas lampung
Computer Science & IT Mathematics

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(α, β)-Derivation on Matrix Ring Mn(R) Dania Azzahra; Fitriani Fitriani; Ahmad Faisol
Euler : Jurnal Ilmiah Matematika, Sains dan Teknologi Volume 14 Issue 1 April 2026
Publisher : Universitas Negeri Gorontalo

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.37905/euler.v14i1.37789

Abstract

In ring theory, a derivation is an additive mapping d:R→R satisfying Leibniz’s rule. A well-known generalization of this notion is the (α,β)-derivation, defined with respect to two ring endomorphisms α and β. In this paper, we study (α,β)-derivations on the matrix ring Mₙ(R) and several of its subrings, including scalar matrices, diagonal matrices, and upper and lower triangular matrix rings. It is shown that an (α,β)-derivation on the base ring RRR induces an (α′,β′)-derivation on these matrix subrings via entrywise extension, preserving their structural properties. Furthermore, we examine certain properties of (α,β)-derivations on the direct product ring R×R. In particular, we show that the sum of two (α,β)-derivations does not necessarily form an (α,β)-derivation, which is demonstrated through a counterexample.
The Connection Between Jordan Derivations and Nilpotent Derivations on 2-Torsion-Free Semiprime Rings Rahmah Mutiara Ayu Ningtiyas; Fitriani Fitriani; Ahmad Faisol
Jambura Journal of Mathematics Vol 8, No 2: August 2025
Publisher : Department of Mathematics, Universitas Negeri Gorontalo

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.37905/jjom.v8i2.37723

Abstract

Let R be a ring. A derivation on a ring is an additive map that satisfies the Leibniz rule d(ab) = d(a)b + ad(b), for every a, b ∈ R. A derivation d on a ring R is called a nilpotent derivation if there exists a natural number n such that dⁿ(R) = 0. This research studies nilpotent Jordan derivations on 2-torsion-free semiprime rings. Motivated by the results of Brešar, Chung, and Luh on Jordan derivations and nilpotent derivations, the study aims to analyze their nilpotency patterns and the behavior of inner derivations generated by nilpotent elements. Using a deductive literature-study approach through mathematical proofs and examples, we show that if d²ⁿ(R) = 0 for some n ∈ N, then d²ⁿ⁻¹(R) = 0. Consequently, every nonzero nilpotent Jordan derivation has an odd nilpotency index. In addition, for a nilpotent element P ∈ R with nilpotency index n, the inner derivation dₚ(x) = [p, x] is nilpotent with nilpotency index at most 2n − 1. These results contribute to a deeper understanding of derivation structures on 2-torsion-free semiprime rings.
Co-Authors Ahmad Faisol Dania Azzahra Pentatito Gunowibowo Rahmah Mutiara Ayu Ningtiyas Sri Hastuti Noer