Some results on the commutativity of prime near-rings using generalized derivations

  • Tadesse Wubshet Woldia University
  • Tilahun Abebaw Addis Ababa University
Keywords: Commutativity; Derivation; 5-torsion free; Generalized derivation; Near-ring; Prime near-ring; 3-torsion free

Abstract

Given a right near ring N, an additive mapping D: N →N is said to be a derivation on N, if D(xy) = D(x)y + xD(y) for all x, y ∈ N and an additive mapping F : N → N satisfying F(xy) = F(x)y + xD(y) for all x, y ∈ N, is called generalized derivation on N associated with the derivation D. The aim of this paper is to study the commutativity of a near-ring using some properties of generalized derivations on the given near-ring and we proved the following results. (a) The commutativity of 3-torsion free prime near-ring N with a generalized derivation F associated with non-zero idempotent derivation D on N satisfying the conditions F2[x, y] - [x, y] = 0 for all x, y ∈ N and F2(xoy) - (xoy) = 0 for all x, y ∈ N and (b) commutativity of 5-torsion free prime near ring N with a generalized derivation F associated with non-zero idempotent derivation D on N satisfying the conditions F2[x, y] + [x, y] = 0 for all x, y ∈ N and F2(xoy) + (xoy) = 0 for all x, y ∈ N are proved in this article. These results may help us to study more about the commutativity of general near-rings.

Author Biographies

Tadesse Wubshet, Woldia University

Department of Mathematics

Tilahun Abebaw, Addis Ababa University

Department of Mathematics

Published
2020-08-23
Section
Research articles

Journal Identifiers


eISSN: 2520-7997
print ISSN: 0379-2897