Commutativity of Near-rings with Derivations

Abstract

In this paper we study some conditions under which a near-ring R admitting a (multiplicative) (σ, τ )-derivation d must be a commutative ring with constrained-suitable conditions on d, σ and τ . Consequently, we obtain some results which generalize some recent theorems in the literature.

On some properties of near-rings

“…where U is a subsemigroup of (R, •) which provides an interesting extension of many known results (see [5,7,9,10]), where we show that U is commutative, V = {u 2 |u ∈ U } is a subsemigroup of R contained in Z (R) and either 2R = {0} or U ⊆ Z (R). Moreover, we study three cases: nU = {0}, nU = {0}, and R is of characteristic zero.…”

“…On the other hand, Bell in [5] has generalized several results of [7] using one (two) sided semigroup ideals of near-rings. Subsequently, many authors, for instance in [8,9] and [10], have investigated various properties of near-rings that make of them commutative rings.…”

On A Class of Idealized Near-Rings Admitting Frobenius Derivations

“…[4,7,8]). Kamal and Khalid [9] in their study of commutativity of near rings with derivations found that any Near-ring admits a derivation if and only if it is zero-symmetric. They also proved some commutativity Theorems for a non-necessarily 3-prime Near-Rings with a suitably constrained derivation d, with the condition that d(a) is not a left zero divisor in R, for some a ∈ R. As a consequence, they attempted to advance further research around the classification of 3-prime Near-Rings admitting derivations.…”

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).…”