Mathematics

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Browsing Mathematics by Author "Abebaw, Tilahun (PhD)"

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    On Commutativity of Prime Near-Rings by Generalized Derivations
    (Addis Ababa University, 2018-06-03) Wubshet, Tadesse; Abebaw, Tilahun (PhD)
    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.