On some properties of near-rings

Abstract

We establish sufficient conditions for 3-prime near-rings to be commutative rings. In particular, for a 3-prime near-ring R with a derivation d, we investigate conditions such as d([U, V]) ⊆ Z(R) , d(U) ⊆ Z(R) , xod(R) ⊆ Z(R) , and Uxo⊆ Z(R). As a by-product, we generalize and extend known results related to rings and near-rings. Furthermore, we discuss the converse of a well-known result in rings and near-rings, namely: if x∈ Z(R) , then d(x) ∈ Z(R). In addition, we provide useful examples illustrating our results.