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

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 𝑢1 , 𝑣1 𝜖 𝑀 and two sided generalized β-derivation G associated with the nonzero two sided β-derivation 𝑔 on M, where 𝛽: 𝑀 → 𝑀 is a homomorphism, satisfying the following conditions: i. 𝐺([𝑝1 , 𝑞1 ]) = 𝑝1 𝑢1[𝛽(𝑝1 ),𝛽(𝑞1 )]𝑝1 𝑣1 ∀ 𝑝1 , 𝑞1 𝜖 𝑀 ii. 𝐺([𝑝1 , 𝑞1 ]) = 𝑝1 𝑢1[𝛽(𝑝1 ),𝛽(𝑞1 )]𝑝1 𝑣1 ∀ 𝑝1 , 𝑞1 𝜖 𝑀 Then M is a commutative ring