Some Commutativity Theorems for Prime Near-rings Involving Derivations

The study depicts that a prime near-ring N is considered to be a commutative ring if there non-negative integers exist i.e., in such a way that N admits a non-zero derivation, where d satisfying one of the conditions like (C1) - (C8). For any x, y ∈ N , we define the following properties

In addition, an example is given to demonstrate the primeness of the hypothesis which is not superfluous. Finally, we can conclude it with some open problems.