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])subseteq Z(R), d(U)subseteq Z(R), x_{o}d(R)subseteq Z(R), and Ux_{o}subseteq 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 xin Z(R), then d(x)in Z(R). In addition, we provide useful examples illustrating our results.