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All Journal Jambura Journal of Mathematics
Rahmah Mutiara Ayu Ningtiyas
Universitas Lampung
Mathematics

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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 Fitriani Fitriani