On Commutativity of Prime Near-Rings by Generalized Derivations
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Date
2018-06-03
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Addis Ababa University
Abstract
Given a nonempty set N together with two binary operations, called addition "+" and
multiplication ".", if (N; +) is a group, (N; :) is a semigroup and multiplication is left or right
distributive over addition, the algebraic structure (N;+; :) is called a near-ring. If (N;+; :) is
a near ring, a function D : N ! N is said to be a derivation on N if D(xy) = D(x)y+xD(y)
for all x; y 2 N and an additive mapping F : N ! N satisfying F(xy) = F(x)y + xD(y)
for all x; y 2 N, is called generalized derivation on N associated with the derivation D.
The aim of this thesis is to study the commutativity of a near-ring using properties of
generalized derivations on the given near ring. If F is a generalized derivation on a near-
ring N associated with a derivation D and if either F[x; y] + [x; y] = 0 for all x; y 2 N;
F[x; y][x; y] = 0 for all x; y 2 N, F(xoy)(xoy) = 0 for all x; y 2 N or F(xoy)+(xoy) =
0 for all x; y 2 N, then it is proved that N is commutative, where [x; y] = xy yx and
xoy = xy+yx for all x; y 2 N: In this thesis, the commutativity of 3-torsion free prime near-
ring N involving generalized derivation F associated with non-zero idempotent derivation D
on N, satisfying
F2[x; y] [x; y] = 0 for all x; y 2 N and F2(xoy) (xoy) = 0 for all x; y 2 N
and commutativity of 5-torsion free prime near ring N involving generalized derivation F
associated with non-zero idempotent derivation D on N, satisfying
F2[x; y] + [x; y] = 0 for all x; y 2 N and F2(xoy) + (xoy) = 0 for all x; y 2 N
are proved. These results can be used to further study the commutativity of prime near-rings
using generalized derivations de_ned on a near-ring.
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Keywords
Commutativity, Prime Near-Rings, Generalized Derivations