ON COMMUTATIVITY OF PRIME NEAR RINGS

In this paper, we prove commutativity of prime near rings by using the notion of β-derivations. Let M be a zero symmetric prime near ring. If there exist p ≥ 0, q ≥ 0 and a nonzero two sided β-derivation d on M, where β : M → M is a homomorphism, such that d satisfy one of the following conditions: [β(s),d(t)] = sp(β(s)oβ(t))sq ∀ s, t ∈ M [β(s),d(t)] = −sp(β(s)oβ(t))sq ∀ s, t ∈ M [d(s),β(t)] = tp(β(s)oβ(t))tq ∀ s, t ∈ M [d(s),β(t)] = −tp(β(s)oβ(t)tq ∀ s, t ∈ M