A NOTE ON COMMUTATIVITY OF PRIME NEAR RING WITH GENERALIZED β-DERIVATION

Abstract

In this paper, we prove commutativity of prime near rings by using the notion of β-derivations. Let M be a prime near ring. If there exist and two sided generalized β-derivation G associated with the non-zero two sided β-derivation on M, where is a homomorphism, satisfying the following conditions: G([p_1,q_1 ])=〖p_1〗^(u_1 ) [β(p_1 ),β(q_1)]〖p_1〗^(v_1 ) ∀ p_1,q_1 ϵ M G([p_1,q_1 ])=〖p_1〗^(u_1 ) [β(p_1 ),β(q_1)]〖p_1〗^(v_1 ) ∀ p_1,q_1 ϵ M Then M is a commutative ring.