Existence of derivations on near-rings

Abstract

ABSTRACT. We obtain conditions on (R, +) which force that the zero map is the only derivation on a zero-symmetric near-ring R. Throughout the paper we construct several new examples of near-rings which are not rings admitting non-zero derivations, non-zero (σ, σ)-derivations and non-zero (1, σ)-derivations.

Commutativity of Near-rings with Derivations

“…Example 3.1 Let R be a 2-distributive near-ring with a distributive element a in R (see [9,Example 2.4] for an example of a 2-distributive near-ring with some distributive elements which is not a distributive near-ring). We will now prove that for any endomorphisms σ,…”

Commutativity of near-rings with (σ;τ)-derivations

“…Example 3.1 Let R be a 2-distributive near-ring with a distributive element a in R (see [9,Example 2.4] for an example of a 2-distributive near-ring with some distributive elements which is not a distributive near-ring). We will now prove that for any endomorphisms σ, τ on…”

Derivation Requirements on Prime Near-Rings for Commutative Rings

“…Suatu hasil kali Lie dan hasil kali Jordan pada near-ring masing -masing didefinisikan sebagai [ , ] = − dan ⋄ = + , dengan , ∈ (Shang, 2011). Penelitian yang berhubungan dengan derivasi pada near-ring terus mengalami perkembangan, yaitu mengenai sifat derivasi pada near-ring, eksistensi derivasi pada nearring serta syarat derivasi near-ring agar membentuk ring komutatif (Argac & Bell, 2001;Gotmare, 2016;Kamal & Al-Shaalan, 2013;Kamal & Al-Shaalan, 2014;Shang, 2011).…”