On Commutativity Of Prime Near-rings By Generalized Derivations

Given a nonempty set N together with two binary operations, called addition "+" andrnmultiplication ".", if (N; +) is a group, (N; :) is a semigroup and multiplication is left or rightrndistributive over addition, the algebraic structure (N;+; :) is called a near-ring. If (N;+; :) isrna near ring, a function D : N À€€! N is said to be a derivation on N if D(xy) = D(x)y+xD(y)rnfor all x; y 2 N and an additive mapping F : N ð€€€! N satisfying F(xy) = F(x)y + xD(y)rnfor all x; y 2 N, is called generalized derivation on N associated with the derivation D.rnThe aim of this thesis is to study the commutativity of a near-ring using properties ofrngeneralized derivations on the given near ring. If F is a generalized derivation on a near-rnring N associated with a derivation D and if either F[x; y] + [x; y] = 0 for all x; y 2 N;rnF[x; y]ð€€€[x; y] = 0 for all x; y 2 N, F(xoy)ð€€€(xoy) = 0 for all x; y 2 N or F(xoy)+(xoy) =rn0 for all x; y 2 N, then it is proved that N is commutative, where [x; y] = xy ð€€€ yx andrnxoy = xy+yx for all x; y 2 N: In this thesis, the commutativity of 3-torsion free prime near-rnring N involving generalized derivation F associated with non-zero idempotent derivation Drnon N, satisfyingrnF2[x; y] ð€€€ [x; y] = 0 for all x; y 2 N and F2(xoy) ð€€€ (xoy) = 0 for all x; y 2 Nrnand commutativity of 5-torsion free prime near ring N involving generalized derivation Frnassociated with non-zero idempotent derivation D on N, satisfyingrnF2[x; y] + [x; y] = 0 for all x; y 2 N and F2(xoy) + (xoy) = 0 for all x; y 2 Nrnare proved. These results can be used to further study the commutativity of prime near-ringsrnusing generalized derivations de_ned on a near-ring.

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