Description
A study on commutativity of prime near-rings via generalized derivation.
Abstract
In this project, we study the commutativity of prime near-rings via generalized derivation and we proved the following results (a) Let N be a 3-torsion free prime near-ring with a generalized derivation F associated with a non-zero idempotent derivation D of N, if F^2[x,y][x,y] for all x,y N then N is commutative. (b) Let N be a 3-torsion free prime near-ring with identity and F a generalized derivation associated with a non-zero idempotent derivation D on N. If k(-1) -k for all nN and F^2 (xoy)-(xoy)0 for all x,yN, then N is commutative. (c) Let N be a 5-torsion free prime near-ring with a generalized derivation F associated with a non-zero idempotent derivation D on N, then N is commutative if F^2[x,y] +[x,y]0 for all x,yN. (d) Let N be a 5-torsion free prime near-ring with identity and F a generalized derivation associated with a non-zero idempotent derivation D on N, then N is commutative if k(-1)-k for all nN and F^2 (xoy)+(xoy)0 for all x,yN..
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